# Properties

 Label 38.7.d.a Level $38$ Weight $7$ Character orbit 38.d Analytic conductor $8.742$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 38.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.74205517755$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 9440 x^{18} + 37579488 x^{16} + 83109028728 x^{14} + 112838424559344 x^{12} + 97808951849137728 x^{10} + 54352282696688610576 x^{8} + 18825433555812367062720 x^{6} + 3765133606401310416173376 x^{4} + 360077534420960567854418432 x^{2} + 8303121574870947536327197696$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{20}\cdot 3^{4}\cdot 19^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{8} q^{2} + ( -2 + \beta_{2} + \beta_{6} ) q^{3} + ( 32 - 32 \beta_{2} ) q^{4} + ( 11 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{12} ) q^{5} + ( 8 - 8 \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{6} + ( -10 + \beta_{1} - \beta_{7} - \beta_{8} ) q^{7} + ( 32 \beta_{7} - 32 \beta_{8} ) q^{8} + ( 221 - 221 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 6 \beta_{6} - 13 \beta_{7} + 25 \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{8} q^{2} + ( -2 + \beta_{2} + \beta_{6} ) q^{3} + ( 32 - 32 \beta_{2} ) q^{4} + ( 11 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{12} ) q^{5} + ( 8 - 8 \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{6} + ( -10 + \beta_{1} - \beta_{7} - \beta_{8} ) q^{7} + ( 32 \beta_{7} - 32 \beta_{8} ) q^{8} + ( 221 - 221 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 6 \beta_{6} - 13 \beta_{7} + 25 \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{9} + ( -\beta_{1} - \beta_{4} + 5 \beta_{5} + \beta_{6} - 13 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} + \beta_{16} + \beta_{19} ) q^{10} + ( -14 + 3 \beta_{3} - \beta_{4} + 8 \beta_{5} - 10 \beta_{6} - 58 \beta_{7} - 59 \beta_{8} - \beta_{9} + \beta_{14} - \beta_{15} - \beta_{18} + \beta_{19} ) q^{11} + ( -32 + 64 \beta_{2} + 32 \beta_{5} + 32 \beta_{6} ) q^{12} + ( 353 + 2 \beta_{1} + 349 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} + 71 \beta_{7} - 4 \beta_{8} - 5 \beta_{9} + \beta_{10} + \beta_{12} - 2 \beta_{15} + 3 \beta_{16} + \beta_{18} - \beta_{19} ) q^{13} + ( 87 - 43 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - 10 \beta_{6} - 5 \beta_{7} + 10 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + 5 \beta_{14} + \beta_{17} - \beta_{18} + 3 \beta_{19} ) q^{14} + ( 476 + \beta_{1} + 476 \beta_{2} + 7 \beta_{3} + 6 \beta_{4} - 18 \beta_{5} + 3 \beta_{6} + 148 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} + 5 \beta_{12} - 4 \beta_{13} - 6 \beta_{15} + 4 \beta_{16} + 5 \beta_{18} + 4 \beta_{19} ) q^{15} -1024 \beta_{2} q^{16} + ( 4 - 9 \beta_{1} - 1167 \beta_{2} + 6 \beta_{3} - 10 \beta_{4} + 24 \beta_{5} + 17 \beta_{6} - 165 \beta_{7} + 92 \beta_{8} + \beta_{9} - 3 \beta_{10} + 14 \beta_{11} + 15 \beta_{12} - 20 \beta_{13} + 4 \beta_{14} + 8 \beta_{15} + 4 \beta_{16} + 2 \beta_{17} + 7 \beta_{18} + 5 \beta_{19} ) q^{17} + ( -362 - \beta_{1} + 734 \beta_{2} + 8 \beta_{3} + 3 \beta_{4} - 44 \beta_{5} - 37 \beta_{6} + 227 \beta_{7} - 226 \beta_{8} + \beta_{9} - 2 \beta_{10} - 6 \beta_{11} + 18 \beta_{12} - \beta_{14} - \beta_{15} + 3 \beta_{16} + 3 \beta_{17} + 6 \beta_{18} + 6 \beta_{19} ) q^{18} + ( 1017 + \beta_{1} - 718 \beta_{2} - 12 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} + 16 \beta_{6} + 241 \beta_{7} - 220 \beta_{8} - 4 \beta_{10} - 10 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 5 \beta_{15} + 4 \beta_{16} + \beta_{17} + 14 \beta_{18} + 5 \beta_{19} ) q^{19} + ( 352 + 32 \beta_{3} - 32 \beta_{5} ) q^{20} + ( 188 - 14 \beta_{1} - 101 \beta_{2} + 27 \beta_{3} - 21 \beta_{4} - 13 \beta_{5} + 25 \beta_{6} - 22 \beta_{7} + 303 \beta_{8} + 9 \beta_{9} + 11 \beta_{10} - 8 \beta_{11} - 19 \beta_{12} - 11 \beta_{13} + 13 \beta_{14} - 2 \beta_{17} - 11 \beta_{18} - 5 \beta_{19} ) q^{21} + ( 3515 + 13 \beta_{1} - 1751 \beta_{2} - 3 \beta_{3} + 10 \beta_{4} + 8 \beta_{5} + 10 \beta_{6} + 15 \beta_{7} + 8 \beta_{8} - 10 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 9 \beta_{13} - 8 \beta_{14} - 4 \beta_{17} + 6 \beta_{18} + 3 \beta_{19} ) q^{22} + ( 834 - 3 \beta_{1} - 833 \beta_{2} - 16 \beta_{3} + 9 \beta_{4} + 114 \beta_{5} + 227 \beta_{6} + 65 \beta_{7} - 116 \beta_{8} - 7 \beta_{9} - 8 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} + 13 \beta_{13} - 18 \beta_{14} - 9 \beta_{15} - 7 \beta_{16} - 14 \beta_{17} + \beta_{18} - 6 \beta_{19} ) q^{23} + ( -256 \beta_{2} + 32 \beta_{4} - 64 \beta_{7} + 32 \beta_{8} - 32 \beta_{11} ) q^{24} + ( -6351 + 6341 \beta_{2} + 25 \beta_{3} - 15 \beta_{4} - 156 \beta_{5} - 181 \beta_{6} + 180 \beta_{7} - 405 \beta_{8} + 5 \beta_{9} + 25 \beta_{10} + 40 \beta_{11} + 76 \beta_{12} - 25 \beta_{13} + 30 \beta_{14} + 15 \beta_{15} + 5 \beta_{16} + 10 \beta_{17} - 15 \beta_{18} - 5 \beta_{19} ) q^{25} + ( -2212 - 4 \beta_{1} - 4 \beta_{3} + 6 \beta_{4} + 22 \beta_{5} + 2 \beta_{6} - 343 \beta_{7} - 360 \beta_{8} - 17 \beta_{9} - 11 \beta_{14} + 11 \beta_{15} - 5 \beta_{16} + 5 \beta_{17} - 7 \beta_{18} + 7 \beta_{19} ) q^{26} + ( 3442 + 24 \beta_{1} - 6954 \beta_{2} - 55 \beta_{3} - 11 \beta_{4} + 5 \beta_{5} - 41 \beta_{6} - 1117 \beta_{7} + 1116 \beta_{8} - \beta_{9} + 2 \beta_{10} + 22 \beta_{11} - 112 \beta_{12} + 46 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - 9 \beta_{16} - 9 \beta_{17} - 36 \beta_{18} - 36 \beta_{19} ) q^{27} + ( -320 + 32 \beta_{1} + 320 \beta_{2} + 64 \beta_{7} - 96 \beta_{8} + 32 \beta_{13} ) q^{28} + ( 4603 + 30 \beta_{1} + 4600 \beta_{2} - 113 \beta_{3} + 3 \beta_{4} + 513 \beta_{5} + 37 \beta_{6} + 279 \beta_{7} - 12 \beta_{8} - 21 \beta_{9} + 18 \beta_{10} - 6 \beta_{11} - 76 \beta_{12} + 9 \beta_{13} - 9 \beta_{15} - 15 \beta_{16} - 12 \beta_{18} - 18 \beta_{19} ) q^{29} + ( -4752 + 5 \beta_{1} + 18 \beta_{3} - 6 \beta_{4} + 168 \beta_{5} - 150 \beta_{6} - 528 \beta_{7} - 492 \beta_{8} + 36 \beta_{9} - 6 \beta_{14} + 6 \beta_{15} - 14 \beta_{16} + 14 \beta_{17} - 21 \beta_{18} + 21 \beta_{19} ) q^{30} + ( -3207 - 74 \beta_{1} + 6438 \beta_{2} - 15 \beta_{3} - 24 \beta_{4} - 379 \beta_{5} - 318 \beta_{6} - 645 \beta_{7} + 671 \beta_{8} + 26 \beta_{9} - 52 \beta_{10} + 48 \beta_{11} + 22 \beta_{12} - 96 \beta_{13} + 25 \beta_{14} + 25 \beta_{15} + 23 \beta_{16} + 23 \beta_{17} + 38 \beta_{18} + 38 \beta_{19} ) q^{31} + 1024 \beta_{7} q^{32} + ( -16744 - 4 \beta_{1} + 8353 \beta_{2} - 146 \beta_{3} + 19 \beta_{4} + 111 \beta_{5} + 34 \beta_{6} + 90 \beta_{7} + 25 \beta_{8} - 13 \beta_{9} - 11 \beta_{10} - 92 \beta_{11} - 16 \beta_{12} + 79 \beta_{13} - 111 \beta_{14} - 2 \beta_{17} + 11 \beta_{18} - 51 \beta_{19} ) q^{33} + ( 2327 - 10 \beta_{1} + 2349 \beta_{2} + 71 \beta_{3} + 34 \beta_{4} - 240 \beta_{5} + 13 \beta_{6} + 1201 \beta_{7} - 23 \beta_{8} + 36 \beta_{9} - 14 \beta_{10} - 31 \beta_{11} + 18 \beta_{12} + 45 \beta_{13} - 28 \beta_{15} - 8 \beta_{16} - 21 \beta_{18} + 14 \beta_{19} ) q^{34} + ( -40 + 66 \beta_{1} - 663 \beta_{2} - 63 \beta_{3} - 26 \beta_{4} - 627 \beta_{5} - 353 \beta_{6} + 747 \beta_{7} - 425 \beta_{8} + 11 \beta_{9} + 21 \beta_{10} - 11 \beta_{11} - 129 \beta_{12} + 146 \beta_{13} - 37 \beta_{14} - 74 \beta_{15} - 64 \beta_{16} - 32 \beta_{17} - 37 \beta_{18} - 29 \beta_{19} ) q^{35} + ( -32 \beta_{1} - 7072 \beta_{2} - 32 \beta_{4} + 160 \beta_{5} + 96 \beta_{6} - 800 \beta_{7} + 416 \beta_{8} + 64 \beta_{11} + 32 \beta_{12} - 32 \beta_{13} + 32 \beta_{14} + 64 \beta_{15} ) q^{36} + ( 2721 + 49 \beta_{1} - 5536 \beta_{2} - 13 \beta_{3} + 6 \beta_{4} + 84 \beta_{5} - 9 \beta_{6} - 4592 \beta_{7} + 4555 \beta_{8} - 37 \beta_{9} + 74 \beta_{10} - 12 \beta_{11} - 100 \beta_{12} + 24 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} - 57 \beta_{16} - 57 \beta_{17} - 84 \beta_{18} - 84 \beta_{19} ) q^{37} + ( -235 - 87 \beta_{1} - 7265 \beta_{2} - 69 \beta_{3} + 26 \beta_{4} + 156 \beta_{5} + 377 \beta_{6} + 641 \beta_{7} - 844 \beta_{8} + 19 \beta_{9} - 4 \beta_{10} + 15 \beta_{11} + 92 \beta_{12} - 75 \beta_{13} + 25 \beta_{14} + 73 \beta_{15} - 15 \beta_{16} + 37 \beta_{17} + 35 \beta_{18} + 30 \beta_{19} ) q^{38} + ( 434 - 68 \beta_{1} + 40 \beta_{3} + 212 \beta_{4} - 745 \beta_{5} + 743 \beta_{6} + 301 \beta_{7} + 215 \beta_{8} - 86 \beta_{9} - \beta_{14} + \beta_{15} - 17 \beta_{16} + 17 \beta_{17} + 87 \beta_{18} - 87 \beta_{19} ) q^{39} + ( 32 \beta_{1} + 32 \beta_{5} - 192 \beta_{6} - 416 \beta_{8} + 32 \beta_{10} - 32 \beta_{11} - 32 \beta_{12} + 32 \beta_{13} - 32 \beta_{14} - 32 \beta_{17} - 32 \beta_{18} ) q^{40} + ( 7192 - 29 \beta_{1} - 3616 \beta_{2} - 92 \beta_{3} + 125 \beta_{4} + 139 \beta_{5} - 1053 \beta_{6} + 25 \beta_{7} - 3099 \beta_{8} + 8 \beta_{9} + 46 \beta_{10} - 14 \beta_{11} - 79 \beta_{12} + 71 \beta_{13} - 139 \beta_{14} - 38 \beta_{17} - 46 \beta_{18} - 32 \beta_{19} ) q^{41} + ( -9344 + 162 \beta_{1} + 9260 \beta_{2} - 100 \beta_{3} + 82 \beta_{4} + 662 \beta_{5} + 841 \beta_{6} + 156 \beta_{7} - 118 \beta_{8} - 10 \beta_{9} + 58 \beta_{10} - 179 \beta_{11} - 222 \beta_{12} + 262 \beta_{13} - 164 \beta_{14} - 82 \beta_{15} - 10 \beta_{16} - 20 \beta_{17} - 76 \beta_{18} - 94 \beta_{19} ) q^{42} + ( 68 - 2 \beta_{1} + 3533 \beta_{2} - 66 \beta_{3} - 189 \beta_{4} - 435 \beta_{5} - 292 \beta_{6} - 280 \beta_{7} + 124 \beta_{8} - 89 \beta_{9} + 86 \beta_{10} + 212 \beta_{11} - 64 \beta_{12} - 60 \beta_{13} + 23 \beta_{14} + 46 \beta_{15} + 6 \beta_{16} + 3 \beta_{17} + 44 \beta_{18} - 21 \beta_{19} ) q^{43} + ( -416 - 96 \beta_{1} + 512 \beta_{2} + 64 \beta_{3} - 32 \beta_{4} - 352 \beta_{5} - 576 \beta_{6} + 1728 \beta_{7} - 3584 \beta_{8} - 32 \beta_{10} + 32 \beta_{11} + 64 \beta_{12} - 160 \beta_{13} + 64 \beta_{14} + 32 \beta_{15} + 64 \beta_{18} + 96 \beta_{19} ) q^{44} + ( 6641 + 55 \beta_{1} + \beta_{3} + 162 \beta_{4} - 1273 \beta_{5} + 1350 \beta_{6} + 4956 \beta_{7} + 5004 \beta_{8} + 48 \beta_{9} + 67 \beta_{14} - 67 \beta_{15} + 3 \beta_{16} - 3 \beta_{17} + 42 \beta_{18} - 42 \beta_{19} ) q^{45} + ( 2566 - 149 \beta_{1} - 5062 \beta_{2} - 108 \beta_{3} + 122 \beta_{4} - 332 \beta_{5} - 410 \beta_{6} + 936 \beta_{7} - 860 \beta_{8} + 76 \beta_{9} - 152 \beta_{10} - 244 \beta_{11} - 64 \beta_{12} - 146 \beta_{13} - 62 \beta_{14} - 62 \beta_{15} + 22 \beta_{16} + 22 \beta_{17} + 111 \beta_{18} + 111 \beta_{19} ) q^{46} + ( -36004 - 187 \beta_{1} + 36168 \beta_{2} + 83 \beta_{3} - 38 \beta_{4} - 208 \beta_{5} - 5 \beta_{6} + 594 \beta_{7} - 1240 \beta_{8} + 72 \beta_{9} - 146 \beta_{10} + 22 \beta_{11} + 351 \beta_{12} - 270 \beta_{13} + 76 \beta_{14} + 38 \beta_{15} + 72 \beta_{16} + 144 \beta_{17} + 191 \beta_{18} + 236 \beta_{19} ) q^{47} + ( 1024 + 1024 \beta_{2} + 1024 \beta_{5} ) q^{48} + ( 13139 + 84 \beta_{1} + 44 \beta_{3} - 326 \beta_{4} + 1289 \beta_{5} - 1559 \beta_{6} + 230 \beta_{7} + 270 \beta_{8} + 40 \beta_{9} + 114 \beta_{14} - 114 \beta_{15} + 45 \beta_{16} - 45 \beta_{17} - 50 \beta_{18} + 50 \beta_{19} ) q^{49} + ( 6170 + 73 \beta_{1} - 12490 \beta_{2} + 20 \beta_{3} - 111 \beta_{4} - 276 \beta_{5} - 133 \beta_{6} - 6213 \beta_{7} + 6219 \beta_{8} + 6 \beta_{9} - 12 \beta_{10} + 222 \beta_{11} + 52 \beta_{12} + 158 \beta_{13} + 56 \beta_{14} + 56 \beta_{15} + 76 \beta_{16} + 76 \beta_{17} - 69 \beta_{18} - 69 \beta_{19} ) q^{50} + ( -16640 + 59 \beta_{1} - 16524 \beta_{2} + 303 \beta_{3} - 304 \beta_{4} + 2226 \beta_{5} - 154 \beta_{6} - 6617 \beta_{7} - 118 \beta_{8} + 173 \beta_{9} - 57 \beta_{10} + 41 \beta_{11} + 5 \beta_{12} + 234 \beta_{13} - 222 \beta_{15} - 59 \beta_{16} - 128 \beta_{18} + 57 \beta_{19} ) q^{51} + ( 22432 + 32 \beta_{1} - 11168 \beta_{2} + 64 \beta_{3} - 64 \beta_{4} - 64 \beta_{5} - 64 \beta_{6} + 32 \beta_{7} + 2144 \beta_{8} - 128 \beta_{9} - 32 \beta_{10} - 32 \beta_{12} + 64 \beta_{14} - 96 \beta_{17} + 32 \beta_{18} - 32 \beta_{19} ) q^{52} + ( -7893 + 58 \beta_{1} - 7870 \beta_{2} - 187 \beta_{3} - 63 \beta_{4} + 525 \beta_{5} + 289 \beta_{6} + 4264 \beta_{7} + 28 \beta_{8} + 114 \beta_{9} - 91 \beta_{10} + 50 \beta_{11} - 120 \beta_{12} - 5 \beta_{13} + 37 \beta_{15} + 68 \beta_{16} - 181 \beta_{18} + 91 \beta_{19} ) q^{53} + ( 79 + 128 \beta_{1} + 37597 \beta_{2} + 237 \beta_{3} - 149 \beta_{4} - 1299 \beta_{5} - 810 \beta_{6} + 7199 \beta_{7} - 3641 \beta_{8} - 132 \beta_{9} + 123 \beta_{10} + 74 \beta_{11} + 109 \beta_{12} - 58 \beta_{13} - 75 \beta_{14} - 150 \beta_{15} + 18 \beta_{16} + 9 \beta_{17} + 17 \beta_{18} - 53 \beta_{19} ) q^{54} + ( 160 - 506 \beta_{1} + 5480 \beta_{2} + 160 \beta_{3} - 234 \beta_{4} + 1738 \beta_{5} + 1226 \beta_{6} - 5027 \beta_{7} + 2820 \beta_{8} + 64 \beta_{9} - 216 \beta_{10} + 516 \beta_{11} + 666 \beta_{12} - 701 \beta_{13} + 282 \beta_{14} + 564 \beta_{15} + 304 \beta_{16} + 152 \beta_{17} + 232 \beta_{18} + 224 \beta_{19} ) q^{55} + ( 1472 - 96 \beta_{1} - 2752 \beta_{2} + 128 \beta_{3} - 64 \beta_{4} - 512 \beta_{5} - 384 \beta_{6} - 416 \beta_{7} + 384 \beta_{8} - 32 \beta_{9} + 64 \beta_{10} + 128 \beta_{11} + 192 \beta_{12} - 256 \beta_{13} + 160 \beta_{14} + 160 \beta_{15} + 32 \beta_{16} + 32 \beta_{17} + 64 \beta_{18} + 64 \beta_{19} ) q^{56} + ( 23224 + 151 \beta_{1} - 15016 \beta_{2} + 446 \beta_{3} + 481 \beta_{4} + 393 \beta_{5} + 2097 \beta_{6} + 4196 \beta_{7} - 8531 \beta_{8} - 323 \beta_{9} + 117 \beta_{10} - 484 \beta_{11} - 70 \beta_{12} - 165 \beta_{13} + 111 \beta_{14} - 16 \beta_{15} - 22 \beta_{16} + 57 \beta_{17} + 3 \beta_{18} - 177 \beta_{19} ) q^{57} + ( -11136 + 72 \beta_{1} - 288 \beta_{3} + 379 \beta_{4} - 54 \beta_{5} + 671 \beta_{6} - 4505 \beta_{7} - 4578 \beta_{8} - 73 \beta_{9} + 121 \beta_{14} - 121 \beta_{15} - 49 \beta_{16} + 49 \beta_{17} + 25 \beta_{18} - 25 \beta_{19} ) q^{58} + ( 87567 + 479 \beta_{1} - 43669 \beta_{2} + 195 \beta_{3} - 35 \beta_{4} + 104 \beta_{5} + 946 \beta_{6} + 86 \beta_{7} - 2805 \beta_{8} - 56 \beta_{9} + 119 \beta_{10} - 139 \beta_{11} - 126 \beta_{12} + 205 \beta_{13} - 104 \beta_{14} - 175 \beta_{17} - 119 \beta_{18} + 173 \beta_{19} ) q^{59} + ( 30304 + 288 \beta_{1} - 15232 \beta_{2} + 192 \beta_{3} - 192 \beta_{5} + 384 \beta_{6} + 4736 \beta_{8} + 128 \beta_{10} + 192 \beta_{11} - 160 \beta_{12} + 128 \beta_{13} + 192 \beta_{14} - 128 \beta_{17} - 128 \beta_{18} - 160 \beta_{19} ) q^{60} + ( 8395 + 124 \beta_{1} - 8337 \beta_{2} + 86 \beta_{3} + 18 \beta_{4} + 1257 \beta_{5} + 1627 \beta_{6} + 6962 \beta_{7} - 14280 \beta_{8} - 122 \beta_{9} + 272 \beta_{10} - 766 \beta_{11} - 103 \beta_{12} + 38 \beta_{13} - 36 \beta_{14} - 18 \beta_{15} - 122 \beta_{16} - 244 \beta_{17} - 168 \beta_{18} - 64 \beta_{19} ) q^{61} + ( -99 + 114 \beta_{1} + 22543 \beta_{2} + 611 \beta_{3} - 431 \beta_{4} - 1219 \beta_{5} - 622 \beta_{6} - 6313 \beta_{7} + 3084 \beta_{8} + 84 \beta_{9} - 37 \beta_{10} + 332 \beta_{11} + 497 \beta_{12} + 276 \beta_{13} - 99 \beta_{14} - 198 \beta_{15} - 94 \beta_{16} - 47 \beta_{17} - 67 \beta_{18} - 15 \beta_{19} ) q^{62} + ( 55410 + 282 \beta_{1} - 55306 \beta_{2} - 10 \beta_{3} + 50 \beta_{4} + 316 \beta_{5} + 882 \beta_{6} + 12610 \beta_{7} - 24904 \beta_{8} - 24 \beta_{10} + 520 \beta_{11} - 220 \beta_{12} + 292 \beta_{13} - 100 \beta_{14} - 50 \beta_{15} + 64 \beta_{18} + 104 \beta_{19} ) q^{63} -32768 q^{64} + ( -3480 - 117 \beta_{1} + 7070 \beta_{2} + 700 \beta_{3} + 824 \beta_{4} - 4381 \beta_{5} - 4163 \beta_{6} - 898 \beta_{7} + 1069 \beta_{8} + 171 \beta_{9} - 342 \beta_{10} - 1648 \beta_{11} + 1742 \beta_{12} + 108 \beta_{13} + 106 \beta_{14} + 106 \beta_{15} + 251 \beta_{16} + 251 \beta_{17} + 226 \beta_{18} + 226 \beta_{19} ) q^{65} + ( -720 - 555 \beta_{1} + 492 \beta_{2} + 28 \beta_{3} - 109 \beta_{4} + 1747 \beta_{5} + 3740 \beta_{6} - 9232 \beta_{7} + 17825 \beta_{8} + 5 \beta_{9} + 61 \beta_{10} + 234 \beta_{11} - 97 \beta_{12} - 583 \beta_{13} + 218 \beta_{14} + 109 \beta_{15} + 5 \beta_{16} + 10 \beta_{17} - 142 \beta_{18} - 223 \beta_{19} ) q^{66} + ( 26745 - 35 \beta_{1} + 26844 \beta_{2} + 1445 \beta_{3} + 35 \beta_{4} + 799 \beta_{5} - 721 \beta_{6} + 14122 \beta_{7} - 35 \beta_{8} + 118 \beta_{9} - 19 \beta_{10} - 121 \beta_{11} + 673 \beta_{12} + 134 \beta_{13} - 207 \beta_{15} - 80 \beta_{16} + 89 \beta_{18} + 19 \beta_{19} ) q^{67} + ( -37376 + 160 \beta_{1} - 192 \beta_{4} + 576 \beta_{5} - 896 \beta_{6} - 2368 \beta_{7} - 2400 \beta_{8} - 32 \beta_{9} - 128 \beta_{14} + 128 \beta_{15} + 64 \beta_{16} - 64 \beta_{17} + 64 \beta_{18} - 64 \beta_{19} ) q^{68} + ( 103107 - 1030 \beta_{1} - 205954 \beta_{2} + 155 \beta_{3} - 654 \beta_{4} - 561 \beta_{5} + 1104 \beta_{6} - 18814 \beta_{7} + 19242 \beta_{8} + 428 \beta_{9} - 856 \beta_{10} + 1308 \beta_{11} + 1166 \beta_{12} - 1204 \beta_{13} + 396 \beta_{14} + 396 \beta_{15} + 236 \beta_{16} + 236 \beta_{17} + 558 \beta_{18} + 558 \beta_{19} ) q^{69} + ( -7919 - 17 \beta_{1} - 7929 \beta_{2} - 1163 \beta_{3} - 607 \beta_{4} - 1353 \beta_{5} + 191 \beta_{6} + 1334 \beta_{7} + 362 \beta_{8} + 27 \beta_{9} - 37 \beta_{10} + 337 \beta_{11} - 387 \beta_{12} - 372 \beta_{13} + 67 \beta_{15} + 47 \beta_{16} + 285 \beta_{18} + 37 \beta_{19} ) q^{70} + ( -58116 + 54 \beta_{1} + 29035 \beta_{2} + 562 \beta_{3} - 749 \beta_{4} + 41 \beta_{5} + 262 \beta_{6} - 94 \beta_{7} + 23032 \beta_{8} + 149 \beta_{9} + 90 \beta_{10} - 790 \beta_{11} - 500 \beta_{12} - 4 \beta_{13} - 41 \beta_{14} + 59 \beta_{17} - 90 \beta_{18} + 103 \beta_{19} ) q^{71} + ( 11808 - 160 \beta_{1} + 11744 \beta_{2} + 416 \beta_{3} + 160 \beta_{4} - 1440 \beta_{5} - 160 \beta_{6} + 7232 \beta_{7} - 64 \beta_{8} - 32 \beta_{9} - 32 \beta_{10} - 96 \beta_{11} + 288 \beta_{12} - 32 \beta_{15} + 96 \beta_{16} + 160 \beta_{18} + 32 \beta_{19} ) q^{72} + ( 111 + 260 \beta_{1} - 21157 \beta_{2} - 260 \beta_{3} - 1626 \beta_{4} - 3110 \beta_{5} - 2111 \beta_{6} + 20796 \beta_{7} - 10665 \beta_{8} - 547 \beta_{9} + 644 \beta_{10} + 1802 \beta_{11} - 520 \beta_{12} + 84 \beta_{13} + 176 \beta_{14} + 352 \beta_{15} - 194 \beta_{16} - 97 \beta_{17} - 228 \beta_{18} - 436 \beta_{19} ) q^{73} + ( 12 + 134 \beta_{1} + 146588 \beta_{2} - 768 \beta_{3} + 163 \beta_{4} + 1694 \beta_{5} + 920 \beta_{6} + 5688 \beta_{7} - 2918 \beta_{8} + 152 \beta_{9} - 222 \beta_{10} - 461 \beta_{11} - 902 \beta_{12} + 230 \beta_{13} - 298 \beta_{14} - 596 \beta_{15} + 140 \beta_{16} + 70 \beta_{17} + 106 \beta_{18} + 164 \beta_{19} ) q^{74} + ( -72076 + 642 \beta_{1} + 144512 \beta_{2} - 649 \beta_{3} + 1141 \beta_{4} - 1312 \beta_{5} - 3204 \beta_{6} - 4312 \beta_{7} + 4261 \beta_{8} - 51 \beta_{9} + 102 \beta_{10} - 2282 \beta_{11} - 1400 \beta_{12} + 1182 \beta_{13} - 931 \beta_{14} - 931 \beta_{15} + 44 \beta_{16} + 44 \beta_{17} + 129 \beta_{18} + 129 \beta_{19} ) q^{75} + ( 9152 - 160 \beta_{1} - 32384 \beta_{2} - 96 \beta_{3} + 160 \beta_{4} + 288 \beta_{5} + 736 \beta_{6} + 7072 \beta_{7} + 448 \beta_{8} - 32 \beta_{10} - 224 \beta_{11} + 416 \beta_{12} + 192 \beta_{13} + 224 \beta_{14} + 64 \beta_{15} + 32 \beta_{16} - 96 \beta_{17} + 64 \beta_{18} - 256 \beta_{19} ) q^{76} + ( -199372 + 302 \beta_{1} - 744 \beta_{3} + 2480 \beta_{4} + 1263 \beta_{5} + 2121 \beta_{6} - 27472 \beta_{7} - 27245 \beta_{8} + 227 \beta_{9} + 77 \beta_{14} - 77 \beta_{15} - 358 \beta_{16} + 358 \beta_{17} - 80 \beta_{18} + 80 \beta_{19} ) q^{77} + ( -5688 - 643 \beta_{1} + 2824 \beta_{2} - 1472 \beta_{3} - 121 \beta_{4} + 429 \beta_{5} - 6528 \beta_{6} - 120 \beta_{7} - 1705 \beta_{8} + 8 \beta_{9} + 29 \beta_{10} - 550 \beta_{11} + 1011 \beta_{12} - 91 \beta_{13} - 429 \beta_{14} - 21 \beta_{17} - 29 \beta_{18} - 32 \beta_{19} ) q^{78} + ( -113796 - 296 \beta_{1} + 57115 \beta_{2} - 208 \beta_{3} + 1537 \beta_{4} + 347 \beta_{5} - 1036 \beta_{6} - 306 \beta_{7} - 12120 \beta_{8} - 11 \beta_{9} + 120 \beta_{10} + 1190 \beta_{11} + 284 \beta_{12} - 186 \beta_{13} - 347 \beta_{14} - 131 \beta_{17} - 120 \beta_{18} + 423 \beta_{19} ) q^{79} + ( 11264 - 11264 \beta_{2} + 1024 \beta_{5} + 1024 \beta_{6} - 1024 \beta_{12} ) q^{80} + ( -344 + 295 \beta_{1} + 230828 \beta_{2} - 1788 \beta_{3} - 1437 \beta_{4} + 5861 \beta_{5} + 2871 \beta_{6} - 9851 \beta_{7} + 4913 \beta_{8} + 168 \beta_{9} + 1318 \beta_{11} - 2083 \beta_{12} + 361 \beta_{13} - 119 \beta_{14} - 238 \beta_{15} - 336 \beta_{16} - 168 \beta_{17} - 352 \beta_{18} - 176 \beta_{19} ) q^{81} + ( 87989 - 109 \beta_{1} - 88453 \beta_{2} - 283 \beta_{3} + 73 \beta_{4} + 213 \beta_{5} - 827 \beta_{6} + 4139 \beta_{7} - 8302 \beta_{8} - 77 \beta_{9} + 121 \beta_{10} + 973 \beta_{11} - 2153 \beta_{12} + 174 \beta_{13} - 146 \beta_{14} - 73 \beta_{15} - 77 \beta_{16} - 154 \beta_{17} - 331 \beta_{18} - 541 \beta_{19} ) q^{82} + ( 175777 - 870 \beta_{1} + 150 \beta_{3} + 1924 \beta_{4} + 2504 \beta_{5} - 552 \beta_{6} + 22659 \beta_{7} + 22523 \beta_{8} - 136 \beta_{9} - 53 \beta_{14} + 53 \beta_{15} - 225 \beta_{16} + 225 \beta_{17} - 89 \beta_{18} + 89 \beta_{19} ) q^{83} + ( 3072 - 6464 \beta_{2} - 256 \beta_{3} + 256 \beta_{4} + 2592 \beta_{5} + 1376 \beta_{6} - 10112 \beta_{7} + 9760 \beta_{8} - 352 \beta_{9} + 704 \beta_{10} - 512 \beta_{11} - 1216 \beta_{12} - 704 \beta_{13} + 416 \beta_{14} + 416 \beta_{15} - 64 \beta_{16} - 64 \beta_{17} - 512 \beta_{18} - 512 \beta_{19} ) q^{84} + ( -917 + 1754 \beta_{1} - 163 \beta_{2} - 600 \beta_{3} + 68 \beta_{4} - 231 \beta_{5} - 2071 \beta_{6} - 27891 \beta_{7} + 57494 \beta_{8} - 313 \beta_{9} + 329 \beta_{10} + 176 \beta_{11} - 1717 \beta_{12} + 2354 \beta_{13} - 136 \beta_{14} - 68 \beta_{15} - 313 \beta_{16} - 626 \beta_{17} - 861 \beta_{18} - 1393 \beta_{19} ) q^{85} + ( 6416 - 187 \beta_{1} + 6384 \beta_{2} - 1176 \beta_{3} - 69 \beta_{4} - 5553 \beta_{5} + 310 \beta_{6} - 1583 \beta_{7} - 493 \beta_{8} - 237 \beta_{9} + 205 \beta_{10} + 64 \beta_{11} - 725 \beta_{12} + 461 \beta_{13} + 59 \beta_{15} - 173 \beta_{16} - 128 \beta_{18} - 205 \beta_{19} ) q^{86} + ( -421618 - 1585 \beta_{1} - 1636 \beta_{3} - 594 \beta_{4} - 3981 \beta_{5} + 4689 \beta_{6} - 8158 \beta_{7} - 8504 \beta_{8} - 346 \beta_{9} - 669 \beta_{14} + 669 \beta_{15} - 43 \beta_{16} + 43 \beta_{17} + 167 \beta_{18} - 167 \beta_{19} ) q^{87} + ( 56128 + 96 \beta_{1} - 112064 \beta_{2} - 256 \beta_{3} - 64 \beta_{4} + 256 \beta_{5} + 448 \beta_{6} - 96 \beta_{7} + 288 \beta_{8} + 192 \beta_{9} - 384 \beta_{10} + 128 \beta_{11} - 128 \beta_{12} + 576 \beta_{13} - 256 \beta_{14} - 256 \beta_{15} - 128 \beta_{16} - 128 \beta_{17} + 288 \beta_{18} + 288 \beta_{19} ) q^{88} + ( 23298 - 166 \beta_{1} + 22725 \beta_{2} + 1222 \beta_{3} - 3736 \beta_{4} - 3282 \beta_{5} - 3307 \beta_{6} - 24115 \beta_{7} - 223 \beta_{8} - 1162 \beta_{9} + 589 \beta_{10} + 2476 \beta_{11} + 869 \beta_{12} - 350 \beta_{13} + 1216 \beta_{15} - 16 \beta_{16} - 111 \beta_{18} - 589 \beta_{19} ) q^{89} + ( -298628 - 104 \beta_{1} + 149164 \beta_{2} - 644 \beta_{3} - 950 \beta_{4} + 76 \beta_{5} - 8300 \beta_{6} - 260 \beta_{7} - 7718 \beta_{8} + 560 \beta_{9} + 116 \beta_{10} - 1026 \beta_{11} + 828 \beta_{12} - 144 \beta_{13} - 76 \beta_{14} + 444 \beta_{17} - 116 \beta_{18} + 260 \beta_{19} ) q^{90} + ( -63246 - 368 \beta_{1} - 63284 \beta_{2} - 3362 \beta_{3} + 3532 \beta_{4} + 6742 \beta_{5} + 3036 \beta_{6} + 19062 \beta_{7} + 1600 \beta_{8} - 12 \beta_{9} - 26 \beta_{10} - 1226 \beta_{11} - 678 \beta_{12} - 1638 \beta_{13} + 1080 \beta_{15} + 64 \beta_{16} + 900 \beta_{18} + 26 \beta_{19} ) q^{91} + ( -448 + 256 \beta_{1} - 26656 \beta_{2} + 320 \beta_{3} - 480 \beta_{4} + 6432 \beta_{5} + 3328 \beta_{6} + 3488 \beta_{7} - 1600 \beta_{8} + 480 \beta_{9} - 256 \beta_{10} + 192 \beta_{11} + 64 \beta_{12} + 416 \beta_{13} - 288 \beta_{14} - 576 \beta_{15} - 448 \beta_{16} - 224 \beta_{17} - 192 \beta_{18} + 32 \beta_{19} ) q^{92} + ( -1043 + 857 \beta_{1} + 354192 \beta_{2} + 995 \beta_{3} - 830 \beta_{4} - 5748 \beta_{5} - 2707 \beta_{6} - 8509 \beta_{7} + 3883 \beta_{8} + 854 \beta_{9} - 501 \beta_{10} + 162 \beta_{11} + 138 \beta_{12} + 1950 \beta_{13} - 668 \beta_{14} - 1336 \beta_{15} - 706 \beta_{16} - 353 \beta_{17} - 879 \beta_{18} - 189 \beta_{19} ) q^{93} + ( 22810 + 193 \beta_{1} - 46018 \beta_{2} + 932 \beta_{3} - 246 \beta_{4} - 428 \beta_{5} + 822 \beta_{6} - 36682 \beta_{7} + 36718 \beta_{8} + 36 \beta_{9} - 72 \beta_{10} + 492 \beta_{11} + 1936 \beta_{12} + 458 \beta_{13} + 606 \beta_{14} + 606 \beta_{15} - 302 \beta_{16} - 302 \beta_{17} - 163 \beta_{18} - 163 \beta_{19} ) q^{94} + ( 34767 + 36 \beta_{1} - 352377 \beta_{2} - 795 \beta_{3} + 1941 \beta_{4} - 7088 \beta_{5} - 9694 \beta_{6} + 41483 \beta_{7} - 29391 \beta_{8} + 1539 \beta_{9} - 854 \beta_{10} - 4806 \beta_{11} - 2588 \beta_{12} + 1116 \beta_{13} - 980 \beta_{14} - 1391 \beta_{15} - 381 \beta_{16} + 100 \beta_{17} - 80 \beta_{18} + 629 \beta_{19} ) q^{95} + ( -8192 + 1024 \beta_{4} + 1024 \beta_{6} - 1024 \beta_{7} - 1024 \beta_{8} ) q^{96} + ( 339849 + 353 \beta_{1} - 170564 \beta_{2} + 2138 \beta_{3} - 1991 \beta_{4} - 319 \beta_{5} + 3068 \beta_{6} + 299 \beta_{7} + 32032 \beta_{8} + 247 \beta_{9} + 234 \beta_{10} - 1672 \beta_{11} - 2851 \beta_{12} + 533 \beta_{13} + 319 \beta_{14} + 13 \beta_{17} - 234 \beta_{18} - 1032 \beta_{19} ) q^{97} + ( -40897 + 556 \beta_{1} + 20677 \beta_{2} + 305 \beta_{3} + 1422 \beta_{4} + 419 \beta_{5} + 9784 \beta_{6} + 579 \beta_{7} - 11173 \beta_{8} + 118 \beta_{9} - 379 \beta_{10} + 1003 \beta_{11} - 149 \beta_{12} + 200 \beta_{13} - 419 \beta_{14} + 497 \beta_{17} + 379 \beta_{18} + 575 \beta_{19} ) q^{98} + ( 62150 - 777 \beta_{1} - 61453 \beta_{2} + 1007 \beta_{3} - 549 \beta_{4} - 17556 \beta_{5} - 28533 \beta_{6} + 15322 \beta_{7} - 31819 \beta_{8} + 414 \beta_{9} - 195 \beta_{10} + 673 \beta_{11} + 5357 \beta_{12} - 1784 \beta_{13} + 1098 \beta_{14} + 549 \beta_{15} + 414 \beta_{16} + 828 \beta_{17} + 653 \beta_{18} + 1111 \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 30q^{3} + 320q^{4} + 112q^{5} + 80q^{6} - 208q^{7} + 2200q^{9} + O(q^{10})$$ $$20q - 30q^{3} + 320q^{4} + 112q^{5} + 80q^{6} - 208q^{7} + 2200q^{9} - 284q^{11} + 10500q^{13} + 1248q^{14} + 14136q^{15} - 10240q^{16} - 11684q^{17} + 12862q^{19} + 7168q^{20} + 2916q^{21} + 52704q^{22} + 8488q^{23} - 2560q^{24} - 63842q^{25} - 43968q^{26} - 3328q^{28} + 137760q^{29} - 94688q^{30} - 250194q^{33} + 70560q^{34} - 6916q^{35} - 70400q^{36} - 77232q^{38} + 9672q^{39} + 109206q^{41} - 92672q^{42} + 35572q^{43} - 4544q^{44} + 131264q^{45} - 361184q^{47} + 30720q^{48} + 259740q^{49} - 496512q^{51} + 336000q^{52} - 236172q^{53} + 375728q^{54} + 56760q^{55} + 314796q^{57} - 225600q^{58} + 1310610q^{59} + 452352q^{60} + 83552q^{61} + 225792q^{62} + 553364q^{63} - 655360q^{64} - 4736q^{66} + 806646q^{67} - 747776q^{68} - 245664q^{70} - 869220q^{71} + 353280q^{72} - 207422q^{73} + 1460832q^{74} - 140096q^{76} - 3988336q^{77} - 85008q^{78} - 1706808q^{79} + 114688q^{80} + 2303170q^{81} + 887712q^{82} + 3527548q^{83} - 5604q^{85} + 195792q^{86} - 8414832q^{87} + 708432q^{89} - 4483104q^{90} - 1914384q^{91} - 271616q^{92} + 3537876q^{93} - 2820356q^{95} - 163840q^{96} + 5113242q^{97} - 612480q^{98} + 603704q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 9440 x^{18} + 37579488 x^{16} + 83109028728 x^{14} + 112838424559344 x^{12} + 97808951849137728 x^{10} + 54352282696688610576 x^{8} + 18825433555812367062720 x^{6} + 3765133606401310416173376 x^{4} + 360077534420960567854418432 x^{2} + 8303121574870947536327197696$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$11\!\cdots\!81$$$$\nu^{18} +$$$$10\!\cdots\!68$$$$\nu^{16} +$$$$40\!\cdots\!12$$$$\nu^{14} +$$$$84\!\cdots\!24$$$$\nu^{12} +$$$$10\!\cdots\!76$$$$\nu^{10} +$$$$80\!\cdots\!56$$$$\nu^{8} +$$$$37\!\cdots\!84$$$$\nu^{6} +$$$$95\!\cdots\!12$$$$\nu^{4} +$$$$11\!\cdots\!12$$$$\nu^{2} +$$$$28\!\cdots\!48$$$$)/$$$$26\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$35\!\cdots\!89$$$$\nu^{19} +$$$$32\!\cdots\!12$$$$\nu^{17} +$$$$12\!\cdots\!48$$$$\nu^{15} +$$$$24\!\cdots\!56$$$$\nu^{13} +$$$$30\!\cdots\!24$$$$\nu^{11} +$$$$21\!\cdots\!24$$$$\nu^{9} +$$$$94\!\cdots\!96$$$$\nu^{7} +$$$$22\!\cdots\!08$$$$\nu^{5} +$$$$24\!\cdots\!08$$$$\nu^{3} +$$$$47\!\cdots\!92$$$$\nu +$$$$36\!\cdots\!00$$$$)/$$$$72\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$92\!\cdots\!83$$$$\nu^{19} +$$$$39\!\cdots\!26$$$$\nu^{18} -$$$$83\!\cdots\!24$$$$\nu^{17} +$$$$35\!\cdots\!48$$$$\nu^{16} -$$$$31\!\cdots\!16$$$$\nu^{15} +$$$$13\!\cdots\!72$$$$\nu^{14} -$$$$64\!\cdots\!32$$$$\nu^{13} +$$$$27\!\cdots\!04$$$$\nu^{12} -$$$$79\!\cdots\!68$$$$\nu^{11} +$$$$33\!\cdots\!76$$$$\nu^{10} -$$$$60\!\cdots\!08$$$$\nu^{9} +$$$$25\!\cdots\!36$$$$\nu^{8} -$$$$27\!\cdots\!12$$$$\nu^{7} +$$$$11\!\cdots\!64$$$$\nu^{6} -$$$$70\!\cdots\!16$$$$\nu^{5} +$$$$29\!\cdots\!32$$$$\nu^{4} -$$$$82\!\cdots\!16$$$$\nu^{3} +$$$$33\!\cdots\!32$$$$\nu^{2} -$$$$20\!\cdots\!64$$$$\nu +$$$$83\!\cdots\!88$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$30\!\cdots\!11$$$$\nu^{19} -$$$$88\!\cdots\!82$$$$\nu^{18} +$$$$27\!\cdots\!08$$$$\nu^{17} -$$$$80\!\cdots\!96$$$$\nu^{16} +$$$$10\!\cdots\!72$$$$\nu^{15} -$$$$30\!\cdots\!64$$$$\nu^{14} +$$$$21\!\cdots\!44$$$$\nu^{13} -$$$$62\!\cdots\!28$$$$\nu^{12} +$$$$26\!\cdots\!56$$$$\nu^{11} -$$$$77\!\cdots\!72$$$$\nu^{10} +$$$$19\!\cdots\!36$$$$\nu^{9} -$$$$58\!\cdots\!32$$$$\nu^{8} +$$$$90\!\cdots\!04$$$$\nu^{7} -$$$$26\!\cdots\!48$$$$\nu^{6} +$$$$23\!\cdots\!72$$$$\nu^{5} -$$$$68\!\cdots\!64$$$$\nu^{4} +$$$$27\!\cdots\!72$$$$\nu^{3} -$$$$81\!\cdots\!64$$$$\nu^{2} +$$$$67\!\cdots\!88$$$$\nu -$$$$19\!\cdots\!56$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!11$$$$\nu^{19} +$$$$67\!\cdots\!74$$$$\nu^{18} -$$$$27\!\cdots\!08$$$$\nu^{17} +$$$$56\!\cdots\!72$$$$\nu^{16} -$$$$10\!\cdots\!72$$$$\nu^{15} +$$$$19\!\cdots\!48$$$$\nu^{14} -$$$$21\!\cdots\!44$$$$\nu^{13} +$$$$37\!\cdots\!96$$$$\nu^{12} -$$$$26\!\cdots\!56$$$$\nu^{11} +$$$$43\!\cdots\!04$$$$\nu^{10} -$$$$19\!\cdots\!36$$$$\nu^{9} +$$$$31\!\cdots\!24$$$$\nu^{8} -$$$$90\!\cdots\!04$$$$\nu^{7} +$$$$14\!\cdots\!36$$$$\nu^{6} -$$$$23\!\cdots\!72$$$$\nu^{5} +$$$$37\!\cdots\!48$$$$\nu^{4} -$$$$27\!\cdots\!72$$$$\nu^{3} +$$$$46\!\cdots\!48$$$$\nu^{2} -$$$$67\!\cdots\!88$$$$\nu +$$$$11\!\cdots\!92$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!11$$$$\nu^{19} -$$$$67\!\cdots\!74$$$$\nu^{18} -$$$$27\!\cdots\!08$$$$\nu^{17} -$$$$56\!\cdots\!72$$$$\nu^{16} -$$$$10\!\cdots\!72$$$$\nu^{15} -$$$$19\!\cdots\!48$$$$\nu^{14} -$$$$21\!\cdots\!44$$$$\nu^{13} -$$$$37\!\cdots\!96$$$$\nu^{12} -$$$$26\!\cdots\!56$$$$\nu^{11} -$$$$43\!\cdots\!04$$$$\nu^{10} -$$$$19\!\cdots\!36$$$$\nu^{9} -$$$$31\!\cdots\!24$$$$\nu^{8} -$$$$90\!\cdots\!04$$$$\nu^{7} -$$$$14\!\cdots\!36$$$$\nu^{6} -$$$$23\!\cdots\!72$$$$\nu^{5} -$$$$37\!\cdots\!48$$$$\nu^{4} -$$$$27\!\cdots\!72$$$$\nu^{3} -$$$$46\!\cdots\!48$$$$\nu^{2} -$$$$67\!\cdots\!88$$$$\nu -$$$$11\!\cdots\!92$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!11$$$$\nu^{19} +$$$$11\!\cdots\!22$$$$\nu^{18} -$$$$27\!\cdots\!08$$$$\nu^{17} +$$$$10\!\cdots\!56$$$$\nu^{16} -$$$$10\!\cdots\!72$$$$\nu^{15} +$$$$38\!\cdots\!84$$$$\nu^{14} -$$$$21\!\cdots\!44$$$$\nu^{13} +$$$$79\!\cdots\!88$$$$\nu^{12} -$$$$26\!\cdots\!56$$$$\nu^{11} +$$$$97\!\cdots\!72$$$$\nu^{10} -$$$$19\!\cdots\!36$$$$\nu^{9} +$$$$73\!\cdots\!92$$$$\nu^{8} -$$$$90\!\cdots\!04$$$$\nu^{7} +$$$$33\!\cdots\!08$$$$\nu^{6} -$$$$23\!\cdots\!72$$$$\nu^{5} +$$$$84\!\cdots\!04$$$$\nu^{4} -$$$$27\!\cdots\!72$$$$\nu^{3} +$$$$98\!\cdots\!04$$$$\nu^{2} -$$$$66\!\cdots\!88$$$$\nu +$$$$24\!\cdots\!36$$$$)/$$$$77\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$30\!\cdots\!11$$$$\nu^{19} +$$$$11\!\cdots\!22$$$$\nu^{18} +$$$$27\!\cdots\!08$$$$\nu^{17} +$$$$10\!\cdots\!56$$$$\nu^{16} +$$$$10\!\cdots\!72$$$$\nu^{15} +$$$$38\!\cdots\!84$$$$\nu^{14} +$$$$21\!\cdots\!44$$$$\nu^{13} +$$$$79\!\cdots\!88$$$$\nu^{12} +$$$$26\!\cdots\!56$$$$\nu^{11} +$$$$97\!\cdots\!72$$$$\nu^{10} +$$$$19\!\cdots\!36$$$$\nu^{9} +$$$$73\!\cdots\!92$$$$\nu^{8} +$$$$90\!\cdots\!04$$$$\nu^{7} +$$$$33\!\cdots\!08$$$$\nu^{6} +$$$$23\!\cdots\!72$$$$\nu^{5} +$$$$84\!\cdots\!04$$$$\nu^{4} +$$$$27\!\cdots\!72$$$$\nu^{3} +$$$$98\!\cdots\!04$$$$\nu^{2} +$$$$66\!\cdots\!88$$$$\nu +$$$$24\!\cdots\!36$$$$)/$$$$77\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$92\!\cdots\!83$$$$\nu^{19} -$$$$30\!\cdots\!70$$$$\nu^{18} -$$$$83\!\cdots\!24$$$$\nu^{17} -$$$$27\!\cdots\!00$$$$\nu^{16} -$$$$31\!\cdots\!16$$$$\nu^{15} -$$$$10\!\cdots\!80$$$$\nu^{14} -$$$$64\!\cdots\!32$$$$\nu^{13} -$$$$21\!\cdots\!80$$$$\nu^{12} -$$$$79\!\cdots\!68$$$$\nu^{11} -$$$$26\!\cdots\!80$$$$\nu^{10} -$$$$60\!\cdots\!08$$$$\nu^{9} -$$$$19\!\cdots\!40$$$$\nu^{8} -$$$$27\!\cdots\!12$$$$\nu^{7} -$$$$88\!\cdots\!80$$$$\nu^{6} -$$$$70\!\cdots\!16$$$$\nu^{5} -$$$$22\!\cdots\!00$$$$\nu^{4} -$$$$82\!\cdots\!16$$$$\nu^{3} -$$$$26\!\cdots\!00$$$$\nu^{2} -$$$$20\!\cdots\!64$$$$\nu -$$$$65\!\cdots\!20$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$97\!\cdots\!95$$$$\nu^{19} -$$$$10\!\cdots\!06$$$$\nu^{18} -$$$$87\!\cdots\!60$$$$\nu^{17} -$$$$91\!\cdots\!68$$$$\nu^{16} -$$$$32\!\cdots\!40$$$$\nu^{15} -$$$$34\!\cdots\!12$$$$\nu^{14} -$$$$66\!\cdots\!80$$$$\nu^{13} -$$$$71\!\cdots\!24$$$$\nu^{12} -$$$$79\!\cdots\!20$$$$\nu^{11} -$$$$87\!\cdots\!76$$$$\nu^{10} -$$$$58\!\cdots\!20$$$$\nu^{9} -$$$$65\!\cdots\!56$$$$\nu^{8} -$$$$26\!\cdots\!80$$$$\nu^{7} -$$$$30\!\cdots\!84$$$$\nu^{6} -$$$$65\!\cdots\!40$$$$\nu^{5} -$$$$76\!\cdots\!12$$$$\nu^{4} -$$$$74\!\cdots\!40$$$$\nu^{3} -$$$$89\!\cdots\!12$$$$\nu^{2} -$$$$18\!\cdots\!60$$$$\nu -$$$$22\!\cdots\!48$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$45\!\cdots\!77$$$$\nu^{19} -$$$$43\!\cdots\!54$$$$\nu^{18} -$$$$41\!\cdots\!16$$$$\nu^{17} -$$$$39\!\cdots\!12$$$$\nu^{16} -$$$$15\!\cdots\!64$$$$\nu^{15} -$$$$15\!\cdots\!08$$$$\nu^{14} -$$$$31\!\cdots\!08$$$$\nu^{13} -$$$$31\!\cdots\!16$$$$\nu^{12} -$$$$38\!\cdots\!32$$$$\nu^{11} -$$$$38\!\cdots\!84$$$$\nu^{10} -$$$$29\!\cdots\!32$$$$\nu^{9} -$$$$29\!\cdots\!04$$$$\nu^{8} -$$$$13\!\cdots\!28$$$$\nu^{7} -$$$$13\!\cdots\!56$$$$\nu^{6} -$$$$33\!\cdots\!44$$$$\nu^{5} -$$$$34\!\cdots\!08$$$$\nu^{4} -$$$$39\!\cdots\!44$$$$\nu^{3} -$$$$40\!\cdots\!08$$$$\nu^{2} -$$$$95\!\cdots\!56$$$$\nu -$$$$99\!\cdots\!32$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!97$$$$\nu^{19} -$$$$32\!\cdots\!92$$$$\nu^{18} -$$$$27\!\cdots\!96$$$$\nu^{17} -$$$$29\!\cdots\!36$$$$\nu^{16} -$$$$10\!\cdots\!24$$$$\nu^{15} -$$$$11\!\cdots\!44$$$$\nu^{14} -$$$$20\!\cdots\!88$$$$\nu^{13} -$$$$22\!\cdots\!68$$$$\nu^{12} -$$$$25\!\cdots\!32$$$$\nu^{11} -$$$$28\!\cdots\!72$$$$\nu^{10} -$$$$19\!\cdots\!12$$$$\nu^{9} -$$$$21\!\cdots\!72$$$$\nu^{8} -$$$$86\!\cdots\!08$$$$\nu^{7} -$$$$95\!\cdots\!88$$$$\nu^{6} -$$$$21\!\cdots\!64$$$$\nu^{5} -$$$$24\!\cdots\!24$$$$\nu^{4} -$$$$25\!\cdots\!64$$$$\nu^{3} -$$$$28\!\cdots\!24$$$$\nu^{2} -$$$$62\!\cdots\!96$$$$\nu -$$$$69\!\cdots\!76$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$10\!\cdots\!97$$$$\nu^{19} -$$$$10\!\cdots\!62$$$$\nu^{18} +$$$$97\!\cdots\!96$$$$\nu^{17} -$$$$95\!\cdots\!36$$$$\nu^{16} +$$$$36\!\cdots\!24$$$$\nu^{15} -$$$$36\!\cdots\!24$$$$\nu^{14} +$$$$76\!\cdots\!88$$$$\nu^{13} -$$$$75\!\cdots\!48$$$$\nu^{12} +$$$$94\!\cdots\!32$$$$\nu^{11} -$$$$94\!\cdots\!52$$$$\nu^{10} +$$$$71\!\cdots\!12$$$$\nu^{9} -$$$$72\!\cdots\!12$$$$\nu^{8} +$$$$32\!\cdots\!08$$$$\nu^{7} -$$$$33\!\cdots\!68$$$$\nu^{6} +$$$$84\!\cdots\!64$$$$\nu^{5} -$$$$86\!\cdots\!24$$$$\nu^{4} +$$$$99\!\cdots\!64$$$$\nu^{3} -$$$$10\!\cdots\!24$$$$\nu^{2} +$$$$24\!\cdots\!96$$$$\nu -$$$$25\!\cdots\!96$$$$)/$$$$47\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$21\!\cdots\!65$$$$\nu^{19} -$$$$49\!\cdots\!42$$$$\nu^{18} +$$$$19\!\cdots\!60$$$$\nu^{17} -$$$$45\!\cdots\!96$$$$\nu^{16} +$$$$73\!\cdots\!20$$$$\nu^{15} -$$$$17\!\cdots\!04$$$$\nu^{14} +$$$$14\!\cdots\!60$$$$\nu^{13} -$$$$35\!\cdots\!68$$$$\nu^{12} +$$$$18\!\cdots\!00$$$$\nu^{11} -$$$$43\!\cdots\!12$$$$\nu^{10} +$$$$13\!\cdots\!60$$$$\nu^{9} -$$$$32\!\cdots\!52$$$$\nu^{8} +$$$$62\!\cdots\!60$$$$\nu^{7} -$$$$14\!\cdots\!88$$$$\nu^{6} +$$$$15\!\cdots\!40$$$$\nu^{5} -$$$$38\!\cdots\!64$$$$\nu^{4} +$$$$18\!\cdots\!40$$$$\nu^{3} -$$$$44\!\cdots\!64$$$$\nu^{2} +$$$$44\!\cdots\!80$$$$\nu -$$$$11\!\cdots\!16$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$21\!\cdots\!65$$$$\nu^{19} +$$$$49\!\cdots\!42$$$$\nu^{18} +$$$$19\!\cdots\!60$$$$\nu^{17} +$$$$45\!\cdots\!96$$$$\nu^{16} +$$$$73\!\cdots\!20$$$$\nu^{15} +$$$$17\!\cdots\!04$$$$\nu^{14} +$$$$14\!\cdots\!60$$$$\nu^{13} +$$$$35\!\cdots\!68$$$$\nu^{12} +$$$$18\!\cdots\!00$$$$\nu^{11} +$$$$43\!\cdots\!12$$$$\nu^{10} +$$$$13\!\cdots\!60$$$$\nu^{9} +$$$$32\!\cdots\!52$$$$\nu^{8} +$$$$62\!\cdots\!60$$$$\nu^{7} +$$$$14\!\cdots\!88$$$$\nu^{6} +$$$$15\!\cdots\!40$$$$\nu^{5} +$$$$38\!\cdots\!64$$$$\nu^{4} +$$$$18\!\cdots\!40$$$$\nu^{3} +$$$$44\!\cdots\!64$$$$\nu^{2} +$$$$44\!\cdots\!80$$$$\nu +$$$$11\!\cdots\!16$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$21\!\cdots\!73$$$$\nu^{19} -$$$$19\!\cdots\!06$$$$\nu^{18} +$$$$19\!\cdots\!24$$$$\nu^{17} -$$$$17\!\cdots\!08$$$$\nu^{16} +$$$$73\!\cdots\!76$$$$\nu^{15} -$$$$67\!\cdots\!52$$$$\nu^{14} +$$$$15\!\cdots\!92$$$$\nu^{13} -$$$$13\!\cdots\!24$$$$\nu^{12} +$$$$18\!\cdots\!28$$$$\nu^{11} -$$$$16\!\cdots\!36$$$$\nu^{10} +$$$$13\!\cdots\!88$$$$\nu^{9} -$$$$12\!\cdots\!76$$$$\nu^{8} +$$$$61\!\cdots\!72$$$$\nu^{7} -$$$$57\!\cdots\!84$$$$\nu^{6} +$$$$15\!\cdots\!16$$$$\nu^{5} -$$$$14\!\cdots\!72$$$$\nu^{4} +$$$$17\!\cdots\!16$$$$\nu^{3} -$$$$16\!\cdots\!72$$$$\nu^{2} +$$$$42\!\cdots\!04$$$$\nu -$$$$41\!\cdots\!08$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$21\!\cdots\!73$$$$\nu^{19} +$$$$19\!\cdots\!06$$$$\nu^{18} +$$$$19\!\cdots\!24$$$$\nu^{17} +$$$$17\!\cdots\!08$$$$\nu^{16} +$$$$73\!\cdots\!76$$$$\nu^{15} +$$$$67\!\cdots\!52$$$$\nu^{14} +$$$$15\!\cdots\!92$$$$\nu^{13} +$$$$13\!\cdots\!24$$$$\nu^{12} +$$$$18\!\cdots\!28$$$$\nu^{11} +$$$$16\!\cdots\!36$$$$\nu^{10} +$$$$13\!\cdots\!88$$$$\nu^{9} +$$$$12\!\cdots\!76$$$$\nu^{8} +$$$$61\!\cdots\!72$$$$\nu^{7} +$$$$57\!\cdots\!84$$$$\nu^{6} +$$$$15\!\cdots\!16$$$$\nu^{5} +$$$$14\!\cdots\!72$$$$\nu^{4} +$$$$17\!\cdots\!16$$$$\nu^{3} +$$$$16\!\cdots\!72$$$$\nu^{2} +$$$$42\!\cdots\!04$$$$\nu +$$$$41\!\cdots\!08$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{18}$$ $$=$$ $$($$$$41\!\cdots\!87$$$$\nu^{19} -$$$$69\!\cdots\!54$$$$\nu^{18} +$$$$37\!\cdots\!96$$$$\nu^{17} -$$$$62\!\cdots\!72$$$$\nu^{16} +$$$$14\!\cdots\!84$$$$\nu^{15} -$$$$23\!\cdots\!68$$$$\nu^{14} +$$$$29\!\cdots\!48$$$$\nu^{13} -$$$$47\!\cdots\!16$$$$\nu^{12} +$$$$35\!\cdots\!92$$$$\nu^{11} -$$$$57\!\cdots\!24$$$$\nu^{10} +$$$$26\!\cdots\!92$$$$\nu^{9} -$$$$42\!\cdots\!84$$$$\nu^{8} +$$$$12\!\cdots\!68$$$$\nu^{7} -$$$$19\!\cdots\!56$$$$\nu^{6} +$$$$31\!\cdots\!64$$$$\nu^{5} -$$$$48\!\cdots\!48$$$$\nu^{4} +$$$$36\!\cdots\!64$$$$\nu^{3} -$$$$56\!\cdots\!48$$$$\nu^{2} +$$$$90\!\cdots\!36$$$$\nu -$$$$13\!\cdots\!72$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$41\!\cdots\!53$$$$\nu^{19} +$$$$69\!\cdots\!54$$$$\nu^{18} +$$$$37\!\cdots\!44$$$$\nu^{17} +$$$$62\!\cdots\!72$$$$\nu^{16} +$$$$14\!\cdots\!16$$$$\nu^{15} +$$$$23\!\cdots\!68$$$$\nu^{14} +$$$$29\!\cdots\!12$$$$\nu^{13} +$$$$47\!\cdots\!16$$$$\nu^{12} +$$$$35\!\cdots\!28$$$$\nu^{11} +$$$$57\!\cdots\!24$$$$\nu^{10} +$$$$27\!\cdots\!08$$$$\nu^{9} +$$$$42\!\cdots\!84$$$$\nu^{8} +$$$$12\!\cdots\!92$$$$\nu^{7} +$$$$19\!\cdots\!56$$$$\nu^{6} +$$$$31\!\cdots\!96$$$$\nu^{5} +$$$$48\!\cdots\!48$$$$\nu^{4} +$$$$37\!\cdots\!96$$$$\nu^{3} +$$$$56\!\cdots\!48$$$$\nu^{2} +$$$$91\!\cdots\!64$$$$\nu +$$$$13\!\cdots\!72$$$$)/$$$$94\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{8} + \beta_{7} - 4 \beta_{6} - 4 \beta_{5}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} - 24 \beta_{8} - 24 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{1} - 1890$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-162 \beta_{19} - 162 \beta_{18} - 45 \beta_{17} - 45 \beta_{16} + 31 \beta_{15} + 31 \beta_{14} + 160 \beta_{13} - 478 \beta_{12} + 166 \beta_{11} + 14 \beta_{10} - 7 \beta_{9} + 7778 \beta_{8} - 7785 \beta_{7} + 5953 \beta_{6} + 6116 \beta_{5} - 83 \beta_{4} - 232 \beta_{3} - 41010 \beta_{2} + 87 \beta_{1} + 20350$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$-394 \beta_{19} + 394 \beta_{18} - 222 \beta_{17} + 222 \beta_{16} - 2168 \beta_{15} + 2168 \beta_{14} - 546 \beta_{9} + 40896 \beta_{8} + 41442 \beta_{7} + 16196 \beta_{6} - 18392 \beta_{5} + 2970 \beta_{4} + 4378 \beta_{3} + 2128 \beta_{1} + 1397350$$ $$\nu^{5}$$ $$=$$ $$($$$$224538 \beta_{19} + 224538 \beta_{18} + 72264 \beta_{17} + 72264 \beta_{16} - 110036 \beta_{15} - 110036 \beta_{14} - 158388 \beta_{13} + 645312 \beta_{12} - 429034 \beta_{11} - 14776 \beta_{10} + 7388 \beta_{9} - 12793006 \beta_{8} + 12800394 \beta_{7} - 5615201 \beta_{6} - 5730728 \beta_{5} + 214517 \beta_{4} + 315268 \beta_{3} + 62638220 \beta_{2} - 86582 \beta_{1} - 31101960$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$1734108 \beta_{19} - 1734108 \beta_{18} + 782295 \beta_{17} - 782295 \beta_{16} + 4558569 \beta_{15} - 4558569 \beta_{14} + 2027031 \beta_{9} - 135754008 \beta_{8} - 137781039 \beta_{7} - 45232359 \beta_{6} + 50972178 \beta_{5} - 9758847 \beta_{4} - 12030450 \beta_{3} - 4216431 \beta_{1} - 2634390176$$ $$\nu^{7}$$ $$=$$ $$-269118981 \beta_{19} - 269118981 \beta_{18} - 96766590 \beta_{17} - 96766590 \beta_{16} + 197237262 \beta_{15} + 197237262 \beta_{14} + 134060982 \beta_{13} - 773917032 \beta_{12} + 730225092 \beta_{11} + 6210120 \beta_{10} - 3105060 \beta_{9} + 17586141755 \beta_{8} - 17589246815 \beta_{7} + 6096907538 \beta_{6} + 6121858568 \beta_{5} - 365112546 \beta_{4} - 383853456 \beta_{3} - 87917377206 \beta_{2} + 70135551 \beta_{1} + 43692674682$$ $$\nu^{8}$$ $$=$$ $$-5547034968 \beta_{19} + 5547034968 \beta_{18} - 2279633436 \beta_{17} + 2279633436 \beta_{16} - 10146897800 \beta_{15} + 10146897800 \beta_{14} - 5986485096 \beta_{9} + 405242973120 \beta_{8} + 411229458216 \beta_{7} + 123723385724 \beta_{6} - 138510776220 \beta_{5} + 27315922120 \beta_{4} + 31009242680 \beta_{3} + 8607441800 \beta_{1} + 5721074436072$$ $$\nu^{9}$$ $$=$$ $$641880228630 \beta_{19} + 641880228630 \beta_{18} + 248109227673 \beta_{17} + 248109227673 \beta_{16} - 591588123395 \beta_{15} - 591588123395 \beta_{14} - 229971762872 \beta_{13} + 1856145510230 \beta_{12} - 2158469869310 \beta_{11} + 14832818378 \beta_{10} - 7416409189 \beta_{9} - 45897517810492 \beta_{8} + 45890101401303 \beta_{7} - 14333748552653 \beta_{6} - 14175169963924 \beta_{5} + 1079234934655 \beta_{4} + 935489164304 \beta_{3} + 241016892003714 \beta_{2} - 107569472247 \beta_{1} - 119859149364038$$ $$\nu^{10}$$ $$=$$ $$15896064313532 \beta_{19} - 15896064313532 \beta_{18} + 6271094686164 \beta_{17} - 6271094686164 \beta_{16} + 23884468067884 \beta_{15} - 23884468067884 \beta_{14} + 16398910387308 \beta_{9} - 1136663408061828 \beta_{8} - 1153062318449136 \beta_{7} - 332563795252768 \beta_{6} + 371594395983784 \beta_{5} - 72304932145824 \beta_{4} - 79543404249776 \beta_{3} - 18493284233192 \beta_{1} - 13449774614960192$$ $$\nu^{11}$$ $$=$$ $$-1561667229487080 \beta_{19} - 1561667229487080 \beta_{18} - 631928697429786 \beta_{17} - 631928697429786 \beta_{16} + 1653369508260902 \beta_{15} + 1653369508260902 \beta_{14} + 422145151929480 \beta_{13} - 4542162536717436 \beta_{12} + 5998858679355808 \beta_{11} - 96479611840244 \beta_{10} + 48239805920122 \beta_{9} + 118025689816382584 \beta_{8} - 117977450010462462 \beta_{7} + 35208130721243540 \beta_{6} + 34431542844004232 \beta_{5} - 2999429339677904 \beta_{4} - 2319321074278840 \beta_{3} - 649444237576030364 \beta_{2} + 162832770044618 \beta_{1} + 323112211752607980$$ $$\nu^{12}$$ $$=$$ $$-43425749169362832 \beta_{19} + 43425749169362832 \beta_{18} - 16819359817761960 \beta_{17} + 16819359817761960 \beta_{16} - 58461923295224664 \beta_{15} + 58461923295224664 \beta_{14} - 43540472956350408 \beta_{9} + 3083433814205733984 \beta_{8} + 3126974287162084392 \beta_{7} + 882686830714302552 \beta_{6} - 986692917381963168 \beta_{5} + 187719979776111480 \beta_{4} + 204874568105046432 \beta_{3} + 41815705955345832 \beta_{1} + 33034132484829445984$$ $$\nu^{13}$$ $$=$$ $$3877188394172103900 \beta_{19} + 3877188394172103900 \beta_{18} + 1614272788012882224 \beta_{17} + 1614272788012882224 \beta_{16} - 4469853056628649296 \beta_{15} - 4469853056628649296 \beta_{14} - 842848865539316232 \beta_{13} + 11331091168071746160 \beta_{12} - 16173538213582650216 \beta_{11} + 348977139786341808 \beta_{10} - 174488569893170904 \beta_{9} - 302848455735455631076 \beta_{8} + 302673967165562460172 \beta_{7} - 88590537002253361204 \beta_{6} - 85994824909604738272 \beta_{5} + 8086769106791325108 \beta_{4} + 5840034153929043984 \beta_{3} + 1727788996552183168728 \beta_{2} - 246935862876487212 \beta_{1} - 859842821312026309560$$ $$\nu^{14}$$ $$=$$ $$115898474237264805984 \beta_{19} - 115898474237264805984 \beta_{18} + 44515640329694348400 \beta_{17} - 44515640329694348400 \beta_{16} + 146605211938884324448 \beta_{15} - 146605211938884324448 \beta_{14} + 114034173853636126752 \beta_{9} - 8212541305561073547984 \beta_{8} - 8326575479414709674736 \beta_{7} - 2323411279874618233000 \beta_{6} + 2600126896163221329552 \beta_{5} - 484685361042679151000 \beta_{4} - 529604028856752635584 \beta_{3} - 98673923194643185888 \beta_{1} - 83114608807722096674976$$ $$\nu^{15}$$ $$=$$ $$-9775605054721072220832 \beta_{19} - 9775605054721072220832 \beta_{18} - 4142078011965669146772 \beta_{17} - 4142078011965669146772 \beta_{16} + 11877504012064130116156 \beta_{15} + 11877504012064130116156 \beta_{14} + 1820578366917583175248 \beta_{13} - 28671428491444354091032 \beta_{12} + 42909588015471593031160 \beta_{11} - 1065985750885388993128 \beta_{10} + 532992875442694496564 \beta_{9} + 778491844527583445542568 \beta_{8} - 777958851652140751046004 \beta_{7} + 225903478405053981191092 \beta_{6} + 218251405767597667224464 \beta_{5} - 21454794007735796515580 \beta_{4} - 14868707121164871542080 \beta_{3} - 4556526159138367010405496 \beta_{2} + 377296308016097091060 \beta_{1} + 2267954481639019738485352$$ $$\nu^{16}$$ $$=$$ $$-305583709658783473863232 \beta_{19} + 305583709658783473863232 \beta_{18} - 116930689596104605331040 \beta_{17} + 116930689596104605331040 \beta_{16} - 372940225582700883616256 \beta_{15} + 372940225582700883616256 \beta_{14} - 296909765404117303117632 \beta_{9} + 21647555759682758162567520 \beta_{8} + 21944465525086875465685152 \beta_{7} + 6083657239356043328403488 \beta_{6} - 6815970044928349826455904 \beta_{5} + 1250857672509361249600128 \beta_{4} + 1372003058764100799926080 \beta_{3} + 240427594518659934684448 \beta_{1} + 211931199689625795821755264$$ $$\nu^{17}$$ $$=$$ $$24909635640701609752296048 \beta_{19} + 24909635640701609752296048 \beta_{18} + 10668272506006082779781424 \beta_{17} + 10668272506006082779781424 \beta_{16} - 31261966390130394840109168 \beta_{15} - 31261966390130394840109168 \beta_{14} - 4185728107354290934977696 \beta_{13} + 73242003781222377142561440 \beta_{12} - 112828382963661298484590640 \beta_{11} + 3021869657437601393563168 \beta_{10} - 1510934828718800696781584 \beta_{9} - 2006270211661296260701075808 \beta_{8} + 2004759276832577460004294224 \beta_{7} - 580376124512002754595741400 \beta_{6} - 559072000092064493227945216 \beta_{5} + 56414191481830649242295320 \beta_{4} + 38131936719329989268062304 \beta_{3} + 11947576253305535544906007360 \beta_{2} - 581929224958344770707264 \beta_{1} - 5947367556183347362003926048$$ $$\nu^{18}$$ $$=$$ $$800409327930953192082544128 \beta_{19} - 800409327930953192082544128 \beta_{18} + 305762593459237936028624016 \beta_{17} - 305762593459237936028624016 \beta_{16} + 956636426723021825635526832 \beta_{15} - 956636426723021825635526832 \beta_{14} + 771232781341693804119906960 \beta_{9} - 56716788234058885637280916416 \beta_{8} - 57488021015400579441400823376 \beta_{7} - 15877612083897382532194276848 \beta_{6} + 17805129169852753215344786016 \beta_{5} - 3231646065423322388517769776 \beta_{4} - 3558344495516786687503190688 \beta_{3} - 599097371989156525920234384 \beta_{1} - 544468374977498187048751613120$$ $$\nu^{19}$$ $$=$$ $$-$$$$63\!\cdots\!28$$$$\beta_{19} -$$$$63\!\cdots\!28$$$$\beta_{18} -$$$$27\!\cdots\!60$$$$\beta_{17} -$$$$27\!\cdots\!60$$$$\beta_{16} +$$$$81\!\cdots\!44$$$$\beta_{15} +$$$$81\!\cdots\!44$$$$\beta_{14} +$$$$10\!\cdots\!72$$$$\beta_{13} -$$$$18\!\cdots\!20$$$$\beta_{12} +$$$$29\!\cdots\!44$$$$\beta_{11} -$$$$82\!\cdots\!20$$$$\beta_{10} +$$$$41\!\cdots\!60$$$$\beta_{9} +$$$$51\!\cdots\!44$$$$\beta_{8} -$$$$51\!\cdots\!84$$$$\beta_{7} +$$$$14\!\cdots\!08$$$$\beta_{6} +$$$$14\!\cdots\!36$$$$\beta_{5} -$$$$14\!\cdots\!72$$$$\beta_{4} -$$$$98\!\cdots\!20$$$$\beta_{3} -$$$$31\!\cdots\!76$$$$\beta_{2} +$$$$90\!\cdots\!76$$$$\beta_{1} +$$$$15\!\cdots\!00$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$1 - \beta_{2}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1
 33.8996i 22.2621i 19.8056i − 25.0158i − 50.9515i 40.1546i 19.6100i 5.69488i − 30.5778i − 34.8817i − 33.8996i − 22.2621i − 19.8056i 25.0158i 50.9515i − 40.1546i − 19.6100i − 5.69488i 30.5778i 34.8817i
−4.89898 2.82843i −32.0827 18.5229i 16.0000 + 27.7128i −25.9363 + 44.9230i 104.782 + 181.487i −644.489 181.019i 321.699 + 557.199i 254.123 146.718i
27.2 −4.89898 2.82843i −22.0043 12.7042i 16.0000 + 27.7128i −85.1965 + 147.565i 71.8656 + 124.475i 473.710 181.019i −41.7080 72.2405i 834.752 481.944i
27.3 −4.89898 2.82843i −19.8769 11.4759i 16.0000 + 27.7128i 99.9842 173.178i 64.9176 + 112.441i 131.431 181.019i −101.107 175.122i −979.641 + 565.596i
27.4 −4.89898 2.82843i 18.9396 + 10.9348i 16.0000 + 27.7128i −55.1543 + 95.5300i −61.8564 107.138i −148.805 181.019i −125.361 217.132i 540.399 312.000i
27.5 −4.89898 2.82843i 41.4005 + 23.9026i 16.0000 + 27.7128i 94.3029 163.337i −135.214 234.197i 72.4666 181.019i 778.168 + 1347.83i −923.976 + 533.458i
27.6 4.89898 + 2.82843i −35.0502 20.2362i 16.0000 + 27.7128i −32.0278 + 55.4737i −114.473 198.274i 487.742 181.019i 454.510 + 787.234i −313.807 + 181.176i
27.7 4.89898 + 2.82843i −17.2580 9.96392i 16.0000 + 27.7128i −17.0310 + 29.4985i −56.3644 97.6261i −265.741 181.019i −165.941 287.418i −166.869 + 96.3417i
27.8 4.89898 + 2.82843i −5.20717 3.00636i 16.0000 + 27.7128i 116.624 201.999i −17.0065 29.4562i −326.637 181.019i −346.424 600.023i 1142.68 659.725i
27.9 4.89898 + 2.82843i 26.2059 + 15.1300i 16.0000 + 27.7128i −88.5587 + 153.388i 85.5881 + 148.243i −258.032 181.019i 93.3327 + 161.657i −867.695 + 500.964i
27.10 4.89898 + 2.82843i 29.9332 + 17.2819i 16.0000 + 27.7128i 48.9935 84.8592i 97.7614 + 169.328i 374.355 181.019i 232.830 + 403.274i 480.036 277.149i
31.1 −4.89898 + 2.82843i −32.0827 + 18.5229i 16.0000 27.7128i −25.9363 44.9230i 104.782 181.487i −644.489 181.019i 321.699 557.199i 254.123 + 146.718i
31.2 −4.89898 + 2.82843i −22.0043 + 12.7042i 16.0000 27.7128i −85.1965 147.565i 71.8656 124.475i 473.710 181.019i −41.7080 + 72.2405i 834.752 + 481.944i
31.3 −4.89898 + 2.82843i −19.8769 + 11.4759i 16.0000 27.7128i 99.9842 + 173.178i 64.9176 112.441i 131.431 181.019i −101.107 + 175.122i −979.641 565.596i
31.4 −4.89898 + 2.82843i 18.9396 10.9348i 16.0000 27.7128i −55.1543 95.5300i −61.8564 + 107.138i −148.805 181.019i −125.361 + 217.132i 540.399 + 312.000i
31.5 −4.89898 + 2.82843i 41.4005 23.9026i 16.0000 27.7128i 94.3029 + 163.337i −135.214 + 234.197i 72.4666 181.019i 778.168 1347.83i −923.976 533.458i
31.6 4.89898 2.82843i −35.0502 + 20.2362i 16.0000 27.7128i −32.0278 55.4737i −114.473 + 198.274i 487.742 181.019i 454.510 787.234i −313.807 181.176i
31.7 4.89898 2.82843i −17.2580 + 9.96392i 16.0000 27.7128i −17.0310 29.4985i −56.3644 + 97.6261i −265.741 181.019i −165.941 + 287.418i −166.869 96.3417i
31.8 4.89898 2.82843i −5.20717 + 3.00636i 16.0000 27.7128i 116.624 + 201.999i −17.0065 + 29.4562i −326.637 181.019i −346.424 + 600.023i 1142.68 + 659.725i
31.9 4.89898 2.82843i 26.2059 15.1300i 16.0000 27.7128i −88.5587 153.388i 85.5881 148.243i −258.032 181.019i 93.3327 161.657i −867.695 500.964i
31.10 4.89898 2.82843i 29.9332 17.2819i 16.0000 27.7128i 48.9935 + 84.8592i 97.7614 169.328i 374.355 181.019i 232.830 403.274i 480.036 + 277.149i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.7.d.a 20
3.b odd 2 1 342.7.m.a 20
19.d odd 6 1 inner 38.7.d.a 20
57.f even 6 1 342.7.m.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.7.d.a 20 1.a even 1 1 trivial
38.7.d.a 20 19.d odd 6 1 inner
342.7.m.a 20 3.b odd 2 1
342.7.m.a 20 57.f even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 32 T^{2} + 1024 T^{4} )^{5}$$
$3$ $$1 + 30 T + 2995 T^{2} + 80850 T^{3} + 3440770 T^{4} + 64781586 T^{5} + 1643186853 T^{6} - 1753414398 T^{7} + 210735384606 T^{8} - 14644273403610 T^{9} + 383169317328477 T^{10} + 10636028302425570 T^{11} + 618534406046852244 T^{12} + 4093736935109190498 T^{13} + 28381100956479588573 T^{14} -$$$$17\!\cdots\!98$$$$T^{15} -$$$$41\!\cdots\!87$$$$T^{16} -$$$$18\!\cdots\!28$$$$T^{17} -$$$$20\!\cdots\!58$$$$T^{18} -$$$$74\!\cdots\!00$$$$T^{19} -$$$$11\!\cdots\!96$$$$T^{20} -$$$$53\!\cdots\!00$$$$T^{21} -$$$$10\!\cdots\!78$$$$T^{22} -$$$$69\!\cdots\!92$$$$T^{23} -$$$$11\!\cdots\!47$$$$T^{24} -$$$$37\!\cdots\!02$$$$T^{25} +$$$$42\!\cdots\!33$$$$T^{26} +$$$$44\!\cdots\!82$$$$T^{27} +$$$$49\!\cdots\!84$$$$T^{28} +$$$$61\!\cdots\!30$$$$T^{29} +$$$$16\!\cdots\!77$$$$T^{30} -$$$$45\!\cdots\!90$$$$T^{31} +$$$$47\!\cdots\!46$$$$T^{32} -$$$$28\!\cdots\!22$$$$T^{33} +$$$$19\!\cdots\!93$$$$T^{34} +$$$$56\!\cdots\!14$$$$T^{35} +$$$$21\!\cdots\!70$$$$T^{36} +$$$$37\!\cdots\!50$$$$T^{37} +$$$$10\!\cdots\!95$$$$T^{38} +$$$$73\!\cdots\!70$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$5$ $$1 - 112 T - 39932 T^{2} + 12139568 T^{3} - 261095854 T^{4} - 330574429340 T^{5} + 58877844589700 T^{6} - 384851486447000 T^{7} - 1302739134446953125 T^{8} +$$$$20\!\cdots\!00$$$$T^{9} -$$$$24\!\cdots\!00$$$$T^{10} -$$$$33\!\cdots\!00$$$$T^{11} +$$$$51\!\cdots\!00$$$$T^{12} -$$$$13\!\cdots\!00$$$$T^{13} -$$$$63\!\cdots\!00$$$$T^{14} +$$$$10\!\cdots\!00$$$$T^{15} -$$$$25\!\cdots\!75$$$$T^{16} -$$$$10\!\cdots\!00$$$$T^{17} +$$$$12\!\cdots\!00$$$$T^{18} +$$$$35\!\cdots\!00$$$$T^{19} -$$$$19\!\cdots\!50$$$$T^{20} +$$$$54\!\cdots\!00$$$$T^{21} +$$$$31\!\cdots\!00$$$$T^{22} -$$$$41\!\cdots\!00$$$$T^{23} -$$$$15\!\cdots\!75$$$$T^{24} +$$$$94\!\cdots\!00$$$$T^{25} -$$$$92\!\cdots\!00$$$$T^{26} -$$$$30\!\cdots\!00$$$$T^{27} +$$$$18\!\cdots\!00$$$$T^{28} -$$$$18\!\cdots\!00$$$$T^{29} -$$$$20\!\cdots\!00$$$$T^{30} +$$$$27\!\cdots\!00$$$$T^{31} -$$$$27\!\cdots\!25$$$$T^{32} -$$$$12\!\cdots\!00$$$$T^{33} +$$$$30\!\cdots\!00$$$$T^{34} -$$$$26\!\cdots\!00$$$$T^{35} -$$$$32\!\cdots\!50$$$$T^{36} +$$$$23\!\cdots\!00$$$$T^{37} -$$$$12\!\cdots\!00$$$$T^{38} -$$$$53\!\cdots\!00$$$$T^{39} +$$$$75\!\cdots\!25$$$$T^{40}$$
$7$ $$( 1 + 104 T + 528718 T^{2} + 63824524 T^{3} + 145021134305 T^{4} + 25246919025412 T^{5} + 27226412889973220 T^{6} + 6494707722522763516 T^{7} +$$$$39\!\cdots\!02$$$$T^{8} +$$$$11\!\cdots\!52$$$$T^{9} +$$$$49\!\cdots\!28$$$$T^{10} +$$$$13\!\cdots\!48$$$$T^{11} +$$$$54\!\cdots\!02$$$$T^{12} +$$$$10\!\cdots\!84$$$$T^{13} +$$$$52\!\cdots\!20$$$$T^{14} +$$$$56\!\cdots\!88$$$$T^{15} +$$$$38\!\cdots\!05$$$$T^{16} +$$$$19\!\cdots\!76$$$$T^{17} +$$$$19\!\cdots\!18$$$$T^{18} +$$$$44\!\cdots\!96$$$$T^{19} +$$$$50\!\cdots\!01$$$$T^{20} )^{2}$$
$11$ $$( 1 + 142 T + 6672981 T^{2} - 2081820698 T^{3} + 26313096523932 T^{4} - 9724295551779526 T^{5} + 81184131041911076199 T^{6} -$$$$28\!\cdots\!02$$$$T^{7} +$$$$19\!\cdots\!87$$$$T^{8} -$$$$66\!\cdots\!76$$$$T^{9} +$$$$37\!\cdots\!00$$$$T^{10} -$$$$11\!\cdots\!36$$$$T^{11} +$$$$59\!\cdots\!27$$$$T^{12} -$$$$15\!\cdots\!62$$$$T^{13} +$$$$79\!\cdots\!59$$$$T^{14} -$$$$16\!\cdots\!26$$$$T^{15} +$$$$81\!\cdots\!52$$$$T^{16} -$$$$11\!\cdots\!58$$$$T^{17} +$$$$64\!\cdots\!61$$$$T^{18} +$$$$24\!\cdots\!22$$$$T^{19} +$$$$30\!\cdots\!01$$$$T^{20} )^{2}$$
$13$ $$1 - 10500 T + 68527912 T^{2} - 333668076000 T^{3} + 1267131856687138 T^{4} - 3930882697800669636 T^{5} +$$$$10\!\cdots\!40$$$$T^{6} -$$$$22\!\cdots\!96$$$$T^{7} +$$$$46\!\cdots\!91$$$$T^{8} -$$$$10\!\cdots\!36$$$$T^{9} +$$$$29\!\cdots\!24$$$$T^{10} -$$$$87\!\cdots\!44$$$$T^{11} +$$$$24\!\cdots\!84$$$$T^{12} -$$$$57\!\cdots\!48$$$$T^{13} +$$$$11\!\cdots\!08$$$$T^{14} -$$$$18\!\cdots\!48$$$$T^{15} +$$$$30\!\cdots\!25$$$$T^{16} -$$$$67\!\cdots\!68$$$$T^{17} +$$$$20\!\cdots\!64$$$$T^{18} -$$$$61\!\cdots\!40$$$$T^{19} +$$$$15\!\cdots\!18$$$$T^{20} -$$$$29\!\cdots\!60$$$$T^{21} +$$$$48\!\cdots\!84$$$$T^{22} -$$$$76\!\cdots\!72$$$$T^{23} +$$$$16\!\cdots\!25$$$$T^{24} -$$$$48\!\cdots\!52$$$$T^{25} +$$$$14\!\cdots\!28$$$$T^{26} -$$$$34\!\cdots\!12$$$$T^{27} +$$$$71\!\cdots\!64$$$$T^{28} -$$$$12\!\cdots\!16$$$$T^{29} +$$$$20\!\cdots\!24$$$$T^{30} -$$$$35\!\cdots\!24$$$$T^{31} +$$$$74\!\cdots\!71$$$$T^{32} -$$$$17\!\cdots\!84$$$$T^{33} +$$$$37\!\cdots\!40$$$$T^{34} -$$$$70\!\cdots\!64$$$$T^{35} +$$$$10\!\cdots\!58$$$$T^{36} -$$$$13\!\cdots\!00$$$$T^{37} +$$$$13\!\cdots\!52$$$$T^{38} -$$$$10\!\cdots\!00$$$$T^{39} +$$$$47\!\cdots\!01$$$$T^{40}$$
$17$ $$1 + 11684 T - 71315420 T^{2} - 870064891288 T^{3} + 7075766595654050 T^{4} + 47867735906434693588 T^{5} -$$$$44\!\cdots\!52$$$$T^{6} -$$$$11\!\cdots\!56$$$$T^{7} +$$$$23\!\cdots\!67$$$$T^{8} -$$$$37\!\cdots\!12$$$$T^{9} -$$$$83\!\cdots\!16$$$$T^{10} +$$$$24\!\cdots\!40$$$$T^{11} +$$$$21\!\cdots\!40$$$$T^{12} -$$$$12\!\cdots\!60$$$$T^{13} -$$$$24\!\cdots\!08$$$$T^{14} +$$$$44\!\cdots\!36$$$$T^{15} -$$$$58\!\cdots\!23$$$$T^{16} -$$$$94\!\cdots\!00$$$$T^{17} +$$$$45\!\cdots\!68$$$$T^{18} +$$$$96\!\cdots\!08$$$$T^{19} -$$$$13\!\cdots\!70$$$$T^{20} +$$$$23\!\cdots\!52$$$$T^{21} +$$$$26\!\cdots\!48$$$$T^{22} -$$$$13\!\cdots\!00$$$$T^{23} -$$$$19\!\cdots\!83$$$$T^{24} +$$$$36\!\cdots\!64$$$$T^{25} -$$$$49\!\cdots\!48$$$$T^{26} -$$$$61\!\cdots\!40$$$$T^{27} +$$$$25\!\cdots\!40$$$$T^{28} +$$$$67\!\cdots\!60$$$$T^{29} -$$$$56\!\cdots\!16$$$$T^{30} -$$$$61\!\cdots\!28$$$$T^{31} +$$$$91\!\cdots\!87$$$$T^{32} -$$$$10\!\cdots\!04$$$$T^{33} -$$$$10\!\cdots\!92$$$$T^{34} +$$$$26\!\cdots\!12$$$$T^{35} +$$$$93\!\cdots\!50$$$$T^{36} -$$$$27\!\cdots\!32$$$$T^{37} -$$$$55\!\cdots\!20$$$$T^{38} +$$$$21\!\cdots\!36$$$$T^{39} +$$$$45\!\cdots\!01$$$$T^{40}$$
$19$ $$1 - 12862 T + 44992741 T^{2} + 547471947190 T^{3} - 8045112741145900 T^{4} + 25850899049043919738 T^{5} +$$$$16\!\cdots\!71$$$$T^{6} -$$$$21\!\cdots\!86$$$$T^{7} +$$$$77\!\cdots\!35$$$$T^{8} +$$$$39\!\cdots\!84$$$$T^{9} -$$$$48\!\cdots\!24$$$$T^{10} +$$$$18\!\cdots\!04$$$$T^{11} +$$$$17\!\cdots\!35$$$$T^{12} -$$$$22\!\cdots\!26$$$$T^{13} +$$$$79\!\cdots\!91$$$$T^{14} +$$$$59\!\cdots\!38$$$$T^{15} -$$$$87\!\cdots\!00$$$$T^{16} +$$$$27\!\cdots\!90$$$$T^{17} +$$$$10\!\cdots\!81$$$$T^{18} -$$$$14\!\cdots\!02$$$$T^{19} +$$$$53\!\cdots\!01$$$$T^{20}$$
$23$ $$1 - 8488 T - 597873998 T^{2} + 5829963927344 T^{3} + 140125585271312960 T^{4} -$$$$15\!\cdots\!88$$$$T^{5} -$$$$16\!\cdots\!88$$$$T^{6} +$$$$21\!\cdots\!52$$$$T^{7} +$$$$19\!\cdots\!33$$$$T^{8} -$$$$15\!\cdots\!40$$$$T^{9} -$$$$40\!\cdots\!36$$$$T^{10} +$$$$57\!\cdots\!96$$$$T^{11} +$$$$86\!\cdots\!02$$$$T^{12} +$$$$39\!\cdots\!84$$$$T^{13} -$$$$16\!\cdots\!26$$$$T^{14} +$$$$21\!\cdots\!72$$$$T^{15} +$$$$29\!\cdots\!33$$$$T^{16} -$$$$11\!\cdots\!96$$$$T^{17} -$$$$38\!\cdots\!52$$$$T^{18} +$$$$11\!\cdots\!28$$$$T^{19} +$$$$47\!\cdots\!98$$$$T^{20} +$$$$16\!\cdots\!92$$$$T^{21} -$$$$84\!\cdots\!92$$$$T^{22} -$$$$38\!\cdots\!24$$$$T^{23} +$$$$14\!\cdots\!53$$$$T^{24} +$$$$15\!\cdots\!28$$$$T^{25} -$$$$17\!\cdots\!86$$$$T^{26} +$$$$61\!\cdots\!36$$$$T^{27} +$$$$20\!\cdots\!62$$$$T^{28} +$$$$19\!\cdots\!64$$$$T^{29} -$$$$20\!\cdots\!36$$$$T^{30} -$$$$11\!\cdots\!60$$$$T^{31} +$$$$21\!\cdots\!93$$$$T^{32} +$$$$34\!\cdots\!88$$$$T^{33} -$$$$41\!\cdots\!08$$$$T^{34} -$$$$56\!\cdots\!12$$$$T^{35} +$$$$74\!\cdots\!60$$$$T^{36} +$$$$45\!\cdots\!76$$$$T^{37} -$$$$69\!\cdots\!38$$$$T^{38} -$$$$14\!\cdots\!92$$$$T^{39} +$$$$25\!\cdots\!01$$$$T^{40}$$
$29$ $$1 - 137760 T + 12416969908 T^{2} - 839100390334080 T^{3} + 46994721297201913834 T^{4} -$$$$22\!\cdots\!96$$$$T^{5} +$$$$96\!\cdots\!92$$$$T^{6} -$$$$37\!\cdots\!28$$$$T^{7} +$$$$13\!\cdots\!91$$$$T^{8} -$$$$44\!\cdots\!44$$$$T^{9} +$$$$14\!\cdots\!24$$$$T^{10} -$$$$43\!\cdots\!80$$$$T^{11} +$$$$12\!\cdots\!04$$$$T^{12} -$$$$36\!\cdots\!00$$$$T^{13} +$$$$10\!\cdots\!44$$$$T^{14} -$$$$28\!\cdots\!84$$$$T^{15} +$$$$75\!\cdots\!81$$$$T^{16} -$$$$19\!\cdots\!76$$$$T^{17} +$$$$50\!\cdots\!48$$$$T^{18} -$$$$12\!\cdots\!88$$$$T^{19} +$$$$30\!\cdots\!18$$$$T^{20} -$$$$74\!\cdots\!48$$$$T^{21} +$$$$17\!\cdots\!68$$$$T^{22} -$$$$41\!\cdots\!36$$$$T^{23} +$$$$94\!\cdots\!61$$$$T^{24} -$$$$21\!\cdots\!84$$$$T^{25} +$$$$45\!\cdots\!24$$$$T^{26} -$$$$97\!\cdots\!00$$$$T^{27} +$$$$20\!\cdots\!44$$$$T^{28} -$$$$40\!\cdots\!80$$$$T^{29} +$$$$78\!\cdots\!24$$$$T^{30} -$$$$14\!\cdots\!24$$$$T^{31} +$$$$26\!\cdots\!31$$$$T^{32} -$$$$43\!\cdots\!08$$$$T^{33} +$$$$67\!\cdots\!52$$$$T^{34} -$$$$93\!\cdots\!96$$$$T^{35} +$$$$11\!\cdots\!14$$$$T^{36} -$$$$12\!\cdots\!80$$$$T^{37} +$$$$10\!\cdots\!88$$$$T^{38} -$$$$71\!\cdots\!60$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$
$31$ $$1 - 11009634428 T^{2} + 59381383689340146838 T^{4} -$$$$20\!\cdots\!08$$$$T^{6} +$$$$54\!\cdots\!33$$$$T^{8} -$$$$11\!\cdots\!64$$$$T^{10} +$$$$18\!\cdots\!64$$$$T^{12} -$$$$26\!\cdots\!92$$$$T^{14} +$$$$32\!\cdots\!46$$$$T^{16} -$$$$34\!\cdots\!60$$$$T^{18} +$$$$33\!\cdots\!56$$$$T^{20} -$$$$27\!\cdots\!60$$$$T^{22} +$$$$20\!\cdots\!66$$$$T^{24} -$$$$13\!\cdots\!52$$$$T^{26} +$$$$72\!\cdots\!24$$$$T^{28} -$$$$33\!\cdots\!64$$$$T^{30} +$$$$13\!\cdots\!13$$$$T^{32} -$$$$39\!\cdots\!68$$$$T^{34} +$$$$87\!\cdots\!78$$$$T^{36} -$$$$12\!\cdots\!48$$$$T^{38} +$$$$91\!\cdots\!01$$$$T^{40}$$
$37$ $$1 - 28489653836 T^{2} +$$$$40\!\cdots\!66$$$$T^{4} -$$$$38\!\cdots\!64$$$$T^{6} +$$$$27\!\cdots\!33$$$$T^{8} -$$$$15\!\cdots\!80$$$$T^{10} +$$$$72\!\cdots\!40$$$$T^{12} -$$$$28\!\cdots\!52$$$$T^{14} +$$$$99\!\cdots\!62$$$$T^{16} -$$$$30\!\cdots\!24$$$$T^{18} +$$$$82\!\cdots\!40$$$$T^{20} -$$$$19\!\cdots\!44$$$$T^{22} +$$$$42\!\cdots\!82$$$$T^{24} -$$$$81\!\cdots\!32$$$$T^{26} +$$$$13\!\cdots\!40$$$$T^{28} -$$$$19\!\cdots\!80$$$$T^{30} +$$$$22\!\cdots\!73$$$$T^{32} -$$$$20\!\cdots\!04$$$$T^{34} +$$$$14\!\cdots\!06$$$$T^{36} -$$$$66\!\cdots\!56$$$$T^{38} +$$$$15\!\cdots\!01$$$$T^{40}$$
$41$ $$1 - 109206 T + 29669841493 T^{2} - 2805996262313286 T^{3} +$$$$43\!\cdots\!78$$$$T^{4} -$$$$38\!\cdots\!86$$$$T^{5} +$$$$42\!\cdots\!15$$$$T^{6} -$$$$36\!\cdots\!18$$$$T^{7} +$$$$31\!\cdots\!50$$$$T^{8} -$$$$26\!\cdots\!74$$$$T^{9} +$$$$18\!\cdots\!15$$$$T^{10} -$$$$14\!\cdots\!62$$$$T^{11} +$$$$87\!\cdots\!72$$$$T^{12} -$$$$63\!\cdots\!22$$$$T^{13} +$$$$33\!\cdots\!59$$$$T^{14} -$$$$20\!\cdots\!66$$$$T^{15} +$$$$98\!\cdots\!69$$$$T^{16} -$$$$38\!\cdots\!88$$$$T^{17} +$$$$23\!\cdots\!22$$$$T^{18} +$$$$73\!\cdots\!52$$$$T^{19} +$$$$69\!\cdots\!40$$$$T^{20} +$$$$35\!\cdots\!32$$$$T^{21} +$$$$53\!\cdots\!82$$$$T^{22} -$$$$41\!\cdots\!48$$$$T^{23} +$$$$50\!\cdots\!09$$$$T^{24} -$$$$49\!\cdots\!66$$$$T^{25} +$$$$37\!\cdots\!19$$$$T^{26} -$$$$34\!\cdots\!82$$$$T^{27} +$$$$22\!\cdots\!12$$$$T^{28} -$$$$18\!\cdots\!82$$$$T^{29} +$$$$10\!\cdots\!15$$$$T^{30} -$$$$73\!\cdots\!34$$$$T^{31} +$$$$41\!\cdots\!50$$$$T^{32} -$$$$23\!\cdots\!78$$$$T^{33} +$$$$12\!\cdots\!15$$$$T^{34} -$$$$54\!\cdots\!86$$$$T^{35} +$$$$29\!\cdots\!98$$$$T^{36} -$$$$89\!\cdots\!66$$$$T^{37} +$$$$44\!\cdots\!53$$$$T^{38} -$$$$78\!\cdots\!66$$$$T^{39} +$$$$34\!\cdots\!01$$$$T^{40}$$
$43$ $$1 - 35572 T - 44740981446 T^{2} + 2696909230114472 T^{3} +$$$$10\!\cdots\!92$$$$T^{4} -$$$$84\!\cdots\!84$$$$T^{5} -$$$$16\!\cdots\!40$$$$T^{6} +$$$$16\!\cdots\!68$$$$T^{7} +$$$$17\!\cdots\!85$$$$T^{8} -$$$$23\!\cdots\!68$$$$T^{9} -$$$$13\!\cdots\!24$$$$T^{10} +$$$$26\!\cdots\!84$$$$T^{11} +$$$$57\!\cdots\!22$$$$T^{12} -$$$$22\!\cdots\!96$$$$T^{13} +$$$$78\!\cdots\!14$$$$T^{14} +$$$$14\!\cdots\!24$$$$T^{15} -$$$$43\!\cdots\!91$$$$T^{16} -$$$$71\!\cdots\!96$$$$T^{17} +$$$$48\!\cdots\!64$$$$T^{18} +$$$$17\!\cdots\!68$$$$T^{19} -$$$$35\!\cdots\!30$$$$T^{20} +$$$$10\!\cdots\!32$$$$T^{21} +$$$$19\!\cdots\!64$$$$T^{22} -$$$$18\!\cdots\!04$$$$T^{23} -$$$$69\!\cdots\!91$$$$T^{24} +$$$$14\!\cdots\!76$$$$T^{25} +$$$$50\!\cdots\!14$$$$T^{26} -$$$$89\!\cdots\!04$$$$T^{27} +$$$$14\!\cdots\!22$$$$T^{28} +$$$$42\!\cdots\!16$$$$T^{29} -$$$$13\!\cdots\!24$$$$T^{30} -$$$$15\!\cdots\!32$$$$T^{31} +$$$$70\!\cdots\!85$$$$T^{32} +$$$$43\!\cdots\!32$$$$T^{33} -$$$$26\!\cdots\!40$$$$T^{34} -$$$$87\!\cdots\!16$$$$T^{35} +$$$$68\!\cdots\!92$$$$T^{36} +$$$$11\!\cdots\!28$$$$T^{37} -$$$$11\!\cdots\!46$$$$T^{38} -$$$$58\!\cdots\!28$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$
$47$ $$1 + 361184 T - 3537913970 T^{2} - 14170984074593752 T^{3} -$$$$36\!\cdots\!72$$$$T^{4} +$$$$35\!\cdots\!36$$$$T^{5} +$$$$64\!\cdots\!52$$$$T^{6} -$$$$75\!\cdots\!92$$$$T^{7} +$$$$31\!\cdots\!53$$$$T^{8} +$$$$13\!\cdots\!56$$$$T^{9} -$$$$39\!\cdots\!36$$$$T^{10} -$$$$21\!\cdots\!52$$$$T^{11} +$$$$11\!\cdots\!22$$$$T^{12} +$$$$28\!\cdots\!64$$$$T^{13} -$$$$24\!\cdots\!14$$$$T^{14} -$$$$29\!\cdots\!72$$$$T^{15} +$$$$40\!\cdots\!41$$$$T^{16} +$$$$23\!\cdots\!32$$$$T^{17} -$$$$55\!\cdots\!52$$$$T^{18} -$$$$93\!\cdots\!80$$$$T^{19} +$$$$65\!\cdots\!54$$$$T^{20} -$$$$10\!\cdots\!20$$$$T^{21} -$$$$64\!\cdots\!32$$$$T^{22} +$$$$29\!\cdots\!48$$$$T^{23} +$$$$54\!\cdots\!21$$$$T^{24} -$$$$43\!\cdots\!28$$$$T^{25} -$$$$38\!\cdots\!94$$$$T^{26} +$$$$47\!\cdots\!76$$$$T^{27} +$$$$21\!\cdots\!42$$$$T^{28} -$$$$42\!\cdots\!88$$$$T^{29} -$$$$84\!\cdots\!36$$$$T^{30} +$$$$31\!\cdots\!24$$$$T^{31} +$$$$76\!\cdots\!73$$$$T^{32} -$$$$20\!\cdots\!88$$$$T^{33} +$$$$18\!\cdots\!12$$$$T^{34} +$$$$11\!\cdots\!64$$$$T^{35} -$$$$12\!\cdots\!12$$$$T^{36} -$$$$50\!\cdots\!68$$$$T^{37} -$$$$13\!\cdots\!70$$$$T^{38} +$$$$15\!\cdots\!96$$$$T^{39} +$$$$44\!\cdots\!01$$$$T^{40}$$
$53$ $$1 + 236172 T + 169761171292 T^{2} + 35701829984187408 T^{3} +$$$$14\!\cdots\!78$$$$T^{4} +$$$$28\!\cdots\!52$$$$T^{5} +$$$$83\!\cdots\!96$$$$T^{6} +$$$$15\!\cdots\!48$$$$T^{7} +$$$$36\!\cdots\!11$$$$T^{8} +$$$$62\!\cdots\!80$$$$T^{9} +$$$$13\!\cdots\!44$$$$T^{10} +$$$$20\!\cdots\!56$$$$T^{11} +$$$$38\!\cdots\!44$$$$T^{12} +$$$$53\!\cdots\!00$$$$T^{13} +$$$$91\!\cdots\!84$$$$T^{14} +$$$$11\!\cdots\!96$$$$T^{15} +$$$$18\!\cdots\!73$$$$T^{16} +$$$$22\!\cdots\!00$$$$T^{17} +$$$$36\!\cdots\!12$$$$T^{18} +$$$$43\!\cdots\!88$$$$T^{19} +$$$$75\!\cdots\!74$$$$T^{20} +$$$$95\!\cdots\!52$$$$T^{21} +$$$$17\!\cdots\!92$$$$T^{22} +$$$$24\!\cdots\!00$$$$T^{23} +$$$$45\!\cdots\!13$$$$T^{24} +$$$$63\!\cdots\!04$$$$T^{25} +$$$$10\!\cdots\!64$$$$T^{26} +$$$$14\!\cdots\!00$$$$T^{27} +$$$$22\!\cdots\!84$$$$T^{28} +$$$$26\!\cdots\!64$$$$T^{29} +$$$$37\!\cdots\!44$$$$T^{30} +$$$$39\!\cdots\!20$$$$T^{31} +$$$$51\!\cdots\!51$$$$T^{32} +$$$$47\!\cdots\!72$$$$T^{33} +$$$$57\!\cdots\!76$$$$T^{34} +$$$$43\!\cdots\!48$$$$T^{35} +$$$$48\!\cdots\!38$$$$T^{36} +$$$$26\!\cdots\!72$$$$T^{37} +$$$$28\!\cdots\!12$$$$T^{38} +$$$$87\!\cdots\!68$$$$T^{39} +$$$$81\!\cdots\!01$$$$T^{40}$$
$59$ $$1 - 1310610 T + 1063192429015 T^{2} - 643019654198022150 T^{3} +$$$$31\!\cdots\!98$$$$T^{4} -$$$$13\!\cdots\!62$$$$T^{5} +$$$$50\!\cdots\!37$$$$T^{6} -$$$$17\!\cdots\!46$$$$T^{7} +$$$$54\!\cdots\!94$$$$T^{8} -$$$$16\!\cdots\!66$$$$T^{9} +$$$$44\!\cdots\!09$$$$T^{10} -$$$$11\!\cdots\!74$$$$T^{11} +$$$$29\!\cdots\!72$$$$T^{12} -$$$$71\!\cdots\!14$$$$T^{13} +$$$$16\!\cdots\!25$$$$T^{14} -$$$$37\!\cdots\!22$$$$T^{15} +$$$$82\!\cdots\!97$$$$T^{16} -$$$$17\!\cdots\!04$$$$T^{17} +$$$$37\!\cdots\!54$$$$T^{18} -$$$$78\!\cdots\!16$$$$T^{19} +$$$$16\!\cdots\!24$$$$T^{20} -$$$$32\!\cdots\!56$$$$T^{21} +$$$$66\!\cdots\!74$$$$T^{22} -$$$$13\!\cdots\!84$$$$T^{23} +$$$$26\!\cdots\!17$$$$T^{24} -$$$$50\!\cdots\!22$$$$T^{25} +$$$$94\!\cdots\!25$$$$T^{26} -$$$$17\!\cdots\!34$$$$T^{27} +$$$$29\!\cdots\!12$$$$T^{28} -$$$$49\!\cdots\!14$$$$T^{29} +$$$$79\!\cdots\!09$$$$T^{30} -$$$$12\!\cdots\!06$$$$T^{31} +$$$$17\!\cdots\!14$$$$T^{32} -$$$$23\!\cdots\!66$$$$T^{33} +$$$$28\!\cdots\!57$$$$T^{34} -$$$$32\!\cdots\!62$$$$T^{35} +$$$$31\!\cdots\!18$$$$T^{36} -$$$$27\!\cdots\!50$$$$T^{37} +$$$$18\!\cdots\!15$$$$T^{38} -$$$$98\!\cdots\!10$$$$T^{39} +$$$$31\!\cdots\!01$$$$T^{40}$$
$61$ $$1 - 83552 T - 268922049252 T^{2} + 9127153116882912 T^{3} +$$$$35\!\cdots\!34$$$$T^{4} +$$$$60\!\cdots\!24$$$$T^{5} -$$$$31\!\cdots\!16$$$$T^{6} -$$$$19\!\cdots\!60$$$$T^{7} +$$$$21\!\cdots\!83$$$$T^{8} +$$$$21\!\cdots\!44$$$$T^{9} -$$$$12\!\cdots\!52$$$$T^{10} -$$$$17\!\cdots\!72$$$$T^{11} +$$$$67\!\cdots\!52$$$$T^{12} +$$$$12\!\cdots\!76$$$$T^{13} -$$$$29\!\cdots\!56$$$$T^{14} -$$$$64\!\cdots\!44$$$$T^{15} +$$$$10\!\cdots\!85$$$$T^{16} +$$$$25\!\cdots\!52$$$$T^{17} -$$$$40\!\cdots\!64$$$$T^{18} -$$$$47\!\cdots\!36$$$$T^{19} +$$$$18\!\cdots\!22$$$$T^{20} -$$$$24\!\cdots\!96$$$$T^{21} -$$$$10\!\cdots\!44$$$$T^{22} +$$$$34\!\cdots\!12$$$$T^{23} +$$$$76\!\cdots\!85$$$$T^{24} -$$$$23\!\cdots\!44$$$$T^{25} -$$$$54\!\cdots\!16$$$$T^{26} +$$$$11\!\cdots\!96$$$$T^{27} +$$$$33\!\cdots\!12$$$$T^{28} -$$$$45\!\cdots\!52$$$$T^{29} -$$$$16\!\cdots\!52$$$$T^{30} +$$$$14\!\cdots\!84$$$$T^{31} +$$$$75\!\cdots\!43$$$$T^{32} -$$$$34\!\cdots\!60$$$$T^{33} -$$$$29\!\cdots\!56$$$$T^{34} +$$$$28\!\cdots\!24$$$$T^{35} +$$$$87\!\cdots\!74$$$$T^{36} +$$$$11\!\cdots\!52$$$$T^{37} -$$$$17\!\cdots\!12$$$$T^{38} -$$$$28\!\cdots\!32$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$67$ $$1 - 806646 T + 938925242311 T^{2} - 582424751039974194 T^{3} +$$$$41\!\cdots\!86$$$$T^{4} -$$$$21\!\cdots\!58$$$$T^{5} +$$$$11\!\cdots\!77$$$$T^{6} -$$$$53\!\cdots\!66$$$$T^{7} +$$$$24\!\cdots\!66$$$$T^{8} -$$$$10\!\cdots\!34$$$$T^{9} +$$$$42\!\cdots\!73$$$$T^{10} -$$$$16\!\cdots\!22$$$$T^{11} +$$$$61\!\cdots\!92$$$$T^{12} -$$$$21\!\cdots\!82$$$$T^{13} +$$$$77\!\cdots\!21$$$$T^{14} -$$$$25\!\cdots\!50$$$$T^{15} +$$$$86\!\cdots\!41$$$$T^{16} -$$$$27\!\cdots\!36$$$$T^{17} +$$$$87\!\cdots\!14$$$$T^{18} -$$$$26\!\cdots\!36$$$$T^{19} +$$$$82\!\cdots\!12$$$$T^{20} -$$$$24\!\cdots\!84$$$$T^{21} +$$$$71\!\cdots\!54$$$$T^{22} -$$$$20\!\cdots\!24$$$$T^{23} +$$$$57\!\cdots\!61$$$$T^{24} -$$$$15\!\cdots\!50$$$$T^{25} +$$$$42\!\cdots\!01$$$$T^{26} -$$$$10\!\cdots\!98$$$$T^{27} +$$$$27\!\cdots\!72$$$$T^{28} -$$$$65\!\cdots\!38$$$$T^{29} +$$$$15\!\cdots\!73$$$$T^{30} -$$$$34\!\cdots\!46$$$$T^{31} +$$$$75\!\cdots\!26$$$$T^{32} -$$$$14\!\cdots\!94$$$$T^{33} +$$$$28\!\cdots\!17$$$$T^{34} -$$$$47\!\cdots\!42$$$$T^{35} +$$$$82\!\cdots\!66$$$$T^{36} -$$$$10\!\cdots\!66$$$$T^{37} +$$$$15\!\cdots\!51$$$$T^{38} -$$$$12\!\cdots\!34$$$$T^{39} +$$$$13\!\cdots\!01$$$$T^{40}$$
$71$ $$1 + 869220 T + 1196037946510 T^{2} + 820708956715606200 T^{3} +$$$$67\!\cdots\!84$$$$T^{4} +$$$$39\!\cdots\!56$$$$T^{5} +$$$$25\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!60$$$$T^{7} +$$$$68\!\cdots\!33$$$$T^{8} +$$$$32\!\cdots\!44$$$$T^{9} +$$$$15\!\cdots\!28$$$$T^{10} +$$$$68\!\cdots\!28$$$$T^{11} +$$$$30\!\cdots\!86$$$$T^{12} +$$$$12\!\cdots\!60$$$$T^{13} +$$$$53\!\cdots\!38$$$$T^{14} +$$$$21\!\cdots\!08$$$$T^{15} +$$$$84\!\cdots\!89$$$$T^{16} +$$$$33\!\cdots\!80$$$$T^{17} +$$$$12\!\cdots\!48$$$$T^{18} +$$$$46\!\cdots\!60$$$$T^{19} +$$$$16\!\cdots\!98$$$$T^{20} +$$$$59\!\cdots\!60$$$$T^{21} +$$$$20\!\cdots\!68$$$$T^{22} +$$$$69\!\cdots\!80$$$$T^{23} +$$$$22\!\cdots\!09$$$$T^{24} +$$$$74\!\cdots\!08$$$$T^{25} +$$$$23\!\cdots\!98$$$$T^{26} +$$$$72\!\cdots\!60$$$$T^{27} +$$$$22\!\cdots\!46$$$$T^{28} +$$$$63\!\cdots\!68$$$$T^{29} +$$$$18\!\cdots\!28$$$$T^{30} +$$$$49\!\cdots\!24$$$$T^{31} +$$$$13\!\cdots\!53$$$$T^{32} +$$$$32\!\cdots\!60$$$$T^{33} +$$$$80\!\cdots\!80$$$$T^{34} +$$$$16\!\cdots\!56$$$$T^{35} +$$$$35\!\cdots\!64$$$$T^{36} +$$$$55\!\cdots\!00$$$$T^{37} +$$$$10\!\cdots\!10$$$$T^{38} +$$$$96\!\cdots\!20$$$$T^{39} +$$$$14\!\cdots\!01$$$$T^{40}$$
$73$ $$1 + 207422 T - 710639910291 T^{2} + 30464971710184310 T^{3} +$$$$30\!\cdots\!90$$$$T^{4} -$$$$61\!\cdots\!82$$$$T^{5} -$$$$78\!\cdots\!49$$$$T^{6} +$$$$29\!\cdots\!26$$$$T^{7} +$$$$11\!\cdots\!54$$$$T^{8} -$$$$78\!\cdots\!18$$$$T^{9} -$$$$40\!\cdots\!25$$$$T^{10} +$$$$13\!\cdots\!14$$$$T^{11} -$$$$25\!\cdots\!80$$$$T^{12} -$$$$11\!\cdots\!78$$$$T^{13} +$$$$69\!\cdots\!99$$$$T^{14} -$$$$23\!\cdots\!50$$$$T^{15} -$$$$93\!\cdots\!67$$$$T^{16} +$$$$20\!\cdots\!56$$$$T^{17} +$$$$73\!\cdots\!82$$$$T^{18} -$$$$16\!\cdots\!48$$$$T^{19} -$$$$51\!\cdots\!32$$$$T^{20} -$$$$24\!\cdots\!72$$$$T^{21} +$$$$16\!\cdots\!22$$$$T^{22} +$$$$69\!\cdots\!64$$$$T^{23} -$$$$48\!\cdots\!47$$$$T^{24} -$$$$18\!\cdots\!50$$$$T^{25} +$$$$84\!\cdots\!39$$$$T^{26} -$$$$20\!\cdots\!62$$$$T^{27} -$$$$70\!\cdots\!80$$$$T^{28} +$$$$54\!\cdots\!26$$$$T^{29} -$$$$25\!\cdots\!25$$$$T^{30} -$$$$75\!\cdots\!02$$$$T^{31} +$$$$17\!\cdots\!34$$$$T^{32} +$$$$65\!\cdots\!94$$$$T^{33} -$$$$25\!\cdots\!09$$$$T^{34} -$$$$30\!\cdots\!18$$$$T^{35} +$$$$23\!\cdots\!90$$$$T^{36} +$$$$34\!\cdots\!90$$$$T^{37} -$$$$12\!\cdots\!71$$$$T^{38} +$$$$54\!\cdots\!98$$$$T^{39} +$$$$39\!\cdots\!01$$$$T^{40}$$
$79$ $$1 + 1706808 T + 2556980606986 T^{2} + 2706854270932071984 T^{3} +$$$$24\!\cdots\!52$$$$T^{4} +$$$$19\!\cdots\!76$$$$T^{5} +$$$$12\!\cdots\!00$$$$T^{6} +$$$$78\!\cdots\!36$$$$T^{7} +$$$$43\!\cdots\!73$$$$T^{8} +$$$$22\!\cdots\!52$$$$T^{9} +$$$$11\!\cdots\!08$$$$T^{10} +$$$$60\!\cdots\!64$$$$T^{11} +$$$$33\!\cdots\!74$$$$T^{12} +$$$$19\!\cdots\!60$$$$T^{13} +$$$$11\!\cdots\!94$$$$T^{14} +$$$$59\!\cdots\!92$$$$T^{15} +$$$$30\!\cdots\!97$$$$T^{16} +$$$$14\!\cdots\!68$$$$T^{17} +$$$$66\!\cdots\!92$$$$T^{18} +$$$$30\!\cdots\!44$$$$T^{19} +$$$$14\!\cdots\!18$$$$T^{20} +$$$$73\!\cdots\!24$$$$T^{21} +$$$$39\!\cdots\!72$$$$T^{22} +$$$$20\!\cdots\!48$$$$T^{23} +$$$$10\!\cdots\!57$$$$T^{24} +$$$$50\!\cdots\!92$$$$T^{25} +$$$$22\!\cdots\!74$$$$T^{26} +$$$$96\!\cdots\!60$$$$T^{27} +$$$$41\!\cdots\!14$$$$T^{28} +$$$$17\!\cdots\!84$$$$T^{29} +$$$$83\!\cdots\!08$$$$T^{30} +$$$$39\!\cdots\!92$$$$T^{31} +$$$$18\!\cdots\!93$$$$T^{32} +$$$$81\!\cdots\!96$$$$T^{33} +$$$$32\!\cdots\!00$$$$T^{34} +$$$$11\!\cdots\!76$$$$T^{35} +$$$$36\!\cdots\!92$$$$T^{36} +$$$$97\!\cdots\!44$$$$T^{37} +$$$$22\!\cdots\!46$$$$T^{38} +$$$$36\!\cdots\!48$$$$T^{39} +$$$$51\!\cdots\!01$$$$T^{40}$$
$83$ $$( 1 - 1763774 T + 3161936397093 T^{2} - 3219192037451900522 T^{3} +$$$$32\!\cdots\!92$$$$T^{4} -$$$$23\!\cdots\!22$$$$T^{5} +$$$$17\!\cdots\!15$$$$T^{6} -$$$$96\!\cdots\!66$$$$T^{7} +$$$$61\!\cdots\!67$$$$T^{8} -$$$$30\!\cdots\!16$$$$T^{9} +$$$$19\!\cdots\!44$$$$T^{10} -$$$$98\!\cdots\!04$$$$T^{11} +$$$$66\!\cdots\!87$$$$T^{12} -$$$$33\!\cdots\!94$$$$T^{13} +$$$$19\!\cdots\!15$$$$T^{14} -$$$$87\!\cdots\!78$$$$T^{15} +$$$$40\!\cdots\!52$$$$T^{16} -$$$$12\!\cdots\!58$$$$T^{17} +$$$$41\!\cdots\!13$$$$T^{18} -$$$$75\!\cdots\!46$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20} )^{2}$$
$89$ $$1 - 708432 T + 2065246836748 T^{2} - 1344571964846393280 T^{3} +$$$$17\!\cdots\!74$$$$T^{4} -$$$$13\!\cdots\!60$$$$T^{5} +$$$$88\!\cdots\!00$$$$T^{6} -$$$$88\!\cdots\!68$$$$T^{7} +$$$$30\!\cdots\!19$$$$T^{8} -$$$$40\!\cdots\!68$$$$T^{9} +$$$$12\!\cdots\!68$$$$T^{10} -$$$$79\!\cdots\!44$$$$T^{11} +$$$$70\!\cdots\!60$$$$T^{12} +$$$$22\!\cdots\!40$$$$T^{13} +$$$$52\!\cdots\!76$$$$T^{14} +$$$$79\!\cdots\!04$$$$T^{15} +$$$$33\!\cdots\!49$$$$T^{16} -$$$$24\!\cdots\!08$$$$T^{17} +$$$$15\!\cdots\!32$$$$T^{18} -$$$$22\!\cdots\!60$$$$T^{19} +$$$$67\!\cdots\!78$$$$T^{20} -$$$$11\!\cdots\!60$$$$T^{21} +$$$$38\!\cdots\!72$$$$T^{22} -$$$$29\!\cdots\!48$$$$T^{23} +$$$$20\!\cdots\!09$$$$T^{24} +$$$$24\!\cdots\!04$$$$T^{25} +$$$$78\!\cdots\!36$$$$T^{26} +$$$$16\!\cdots\!40$$$$T^{27} +$$$$26\!\cdots\!60$$$$T^{28} -$$$$14\!\cdots\!04$$$$T^{29} +$$$$11\!\cdots\!68$$$$T^{30} -$$$$18\!\cdots\!48$$$$T^{31} +$$$$69\!\cdots\!99$$$$T^{32} -$$$$10\!\cdots\!08$$$$T^{33} +$$$$49\!\cdots\!00$$$$T^{34} -$$$$37\!\cdots\!60$$$$T^{35} +$$$$24\!\cdots\!14$$$$T^{36} -$$$$92\!\cdots\!80$$$$T^{37} +$$$$70\!\cdots\!88$$$$T^{38} -$$$$12\!\cdots\!12$$$$T^{39} +$$$$84\!\cdots\!01$$$$T^{40}$$
$97$ $$1 - 5113242 T + 16536828207181 T^{2} - 39994485053868801306 T^{3} +$$$$79\!\cdots\!42$$$$T^{4} -$$$$13\!\cdots\!18$$$$T^{5} +$$$$20\!\cdots\!39$$$$T^{6} -$$$$26\!\cdots\!82$$$$T^{7} +$$$$31\!\cdots\!58$$$$T^{8} -$$$$33\!\cdots\!86$$$$T^{9} +$$$$30\!\cdots\!51$$$$T^{10} -$$$$24\!\cdots\!94$$$$T^{11} +$$$$17\!\cdots\!16$$$$T^{12} -$$$$96\!\cdots\!50$$$$T^{13} +$$$$41\!\cdots\!31$$$$T^{14} -$$$$18\!\cdots\!54$$$$T^{15} +$$$$26\!\cdots\!25$$$$T^{16} -$$$$55\!\cdots\!88$$$$T^{17} +$$$$86\!\cdots\!10$$$$T^{18} -$$$$10\!\cdots\!60$$$$T^{19} +$$$$10\!\cdots\!28$$$$T^{20} -$$$$88\!\cdots\!40$$$$T^{21} +$$$$60\!\cdots\!10$$$$T^{22} -$$$$31\!\cdots\!32$$$$T^{23} +$$$$12\!\cdots\!25$$$$T^{24} -$$$$74\!\cdots\!46$$$$T^{25} +$$$$13\!\cdots\!51$$$$T^{26} -$$$$26\!\cdots\!50$$$$T^{27} +$$$$39\!\cdots\!76$$$$T^{28} -$$$$48\!\cdots\!86$$$$T^{29} +$$$$49\!\cdots\!51$$$$T^{30} -$$$$44\!\cdots\!94$$$$T^{31} +$$$$35\!\cdots\!78$$$$T^{32} -$$$$24\!\cdots\!98$$$$T^{33} +$$$$15\!\cdots\!59$$$$T^{34} -$$$$87\!\cdots\!82$$$$T^{35} +$$$$42\!\cdots\!82$$$$T^{36} -$$$$17\!\cdots\!54$$$$T^{37} +$$$$61\!\cdots\!41$$$$T^{38} -$$$$15\!\cdots\!98$$$$T^{39} +$$$$25\!\cdots\!01$$$$T^{40}$$